ML24095A304
| ML24095A304 | |
| Person / Time | |
|---|---|
| Issue date: | 04/29/2024 |
| From: | Pillai R, Wendy Reed, Savara A NRC/RES/DE, Oak Ridge |
| To: | |
| References | |
| TLR–RES/DE/REB?2024?02 | |
| Download: ML24095A304 (1) | |
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1 Technical Letter Report TLR-RES/DE/REB202402 PRELIMINARY ASSESSMENT OF MODELS FOR GENERATING PREDICTIONS OF LONG-TERM CORROSION IN MOLTEN SALTS Date:
April 2024 Prepared by:
Rishi Pillai Oak Ridge National Laboratory Aditya Savara U.S. Nuclear Regulatory Commission Project Manager:
Wendy Reed Senior Physical Scientist Reactor Engineering Branch Division of Engineering Office of Nuclear Regulatory Research U.S. Nuclear Regulatory Commission Washington, DC 20555-0001
2 Technical Letter Report TLR-RES/DE/REB202402 PRELIMINARY ASSESSMENT OF MODELS FOR GENERATING PREDICTIONS OF LONG-TERM CORROSION IN MOLTEN SALTS Date:
April 2024 Prepared in response to Task 9 in NRC Technical Assistance Pertaining to Advanced Reactors in the Areas of Corrosion Experiment Methodology and Evaluation (Accession No. ML24095A304), by:
Rishi Pillai Oak Ridge National Laboratory Aditya Savara U.S. Nuclear Regulatory Commission Project Manager:
Wendy Reed Senior Physical Scientist Reactor Engineering Branch Division of Engineering Office of Nuclear Regulatory Research U.S. Nuclear Regulatory Commission Washington, DC 20555-0001
3 DISCLAIMER: This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any employee, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for any third party's use, or the results of such use, of any information, apparatus, product, or process disclosed in this publication, or represents that its use by such third party complies with applicable law.
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6 ORNL/SPR2023/3116 Materials Science and Technology Division and Chemical Sciences Division PRELIMINARY ASSESSMENT OF MODELS FOR GENERATING PREDICTIONS OF LONG-TERM CORROSION IN MOLTEN SALTS (TASK 9 OF EVALUATING STATIC ISOTHERMAL MOLTEN SALT COMPATIBILITY WITH STRUCTURAL ALLOYS)
Aditya Savara and Rishi Pillai April 2024 Prepared by OAK RIDGE NATIONAL LABORATORY Oak Ridge, TN 37831 managed by UTBATTELLE LLC for the US DEPARTMENT OF ENERGY under contract DEAC0500OR22725
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8 Table of Contents Table of Contents.......................................................................................................................................... 8 Executive Summary....................................................................................................................................... 9 Acronyms Used........................................................................................................................................... 11
- 1. Introduction............................................................................................................................................ 12
- 2. Background and State-of-The-Art........................................................................................................... 14 2.1 AI/ML/DS Augmented Physicochemical Modeling........................................................................... 14 2.2 Corrosion Prediction in Molten Salts................................................................................................ 17
- 3. Current Modeling Challenges and Limitations........................................................................................ 24
- 4. Opportunities for New ML Tools to Augment MSR Long-Term Corrosion Rate Predictions.................. 26 4.1 ML Surrogates for Predicting Metal Alloys Elemental Activities (Demonstration)........................... 27 4.2 ML Prediction of Metal Alloys Atomic-Scale Interatomic Potential Energy and of Elemental Diffusion Activation Energy..................................................................................................................... 33 4.3 ML Descriptor Identification and ML Prediction of Corrosion Rates................................................ 34 4.4 Integration of Atomic-Scale ML Predictions into Multi-scale Modeling Methods........................... 37 4.5 ML Augmented Continuum-Scale Modeling (Fluid Dynamics and Heat Transfer)........................... 39 5 Key Observations..................................................................................................................................... 39 5.1 Long-Term Intrinsic Corrosion Rates Differ from Short-Term Intrinsic Corrosion Rates.................. 40 5.2 Acquisition of Relevant Experimental Data for Databases............................................................... 40 5.3 Need for Combined Efforts Between Modeling and Experimental Studies..................................... 41 5.4 Increasing Focus on Sources of Uncertainty/Variation/etc.............................................................. 41 5.5 ML Augmentation Should Be Gradually Incorporated into Predictions........................................... 42
- 6. Summary................................................................................................................................................. 42 Appendixes.................................................................................................................................................. 44 Appendix 1: Explanation of Various AI/ML/DS Methods........................................................................ 44 Appendix 2: Explainable / Interpretable AI............................................................................................. 46 Appendix 3: Details of Piecewise Gaussian Process Based ML Surrogate Model................................... 49 References.................................................................................................................................................. 52
9 Executive Summary There has been a surge of interest in MSRs in the last ten years, partly based on the Generation IV International Forum (GIF) having included molten salt based nuclear reactors under consideration for the next generation of nuclear reactors.1 Accordingly, molten salt fueled and molten salt cooled reactor concepts are currently being pursued by industry. The current state of knowledge provides sufficient confidence that corrosion resistant technologies exist, such that appropriate structural material and salt choices should enable safe use of MSRs. Although such corrosion resistant technologies exist, there would be a benefit from having improved models to predict the long-term corrosion rates.
Todays experiments, modeling, and monitoring occur for short timescales (minutes, days, and even months) and are not by themselves capable of providing the information needed for precise quantitative prediction of long-term materials degradation (years-scale). Quantitative prediction of the long-term corrosion rates of alloys in molten salts using short-term data requires information and knowledge about changes in intrinsic corrosion rates. It is presently believed that the primary factors for the changes in the intrinsic corrosion rate are due to (a) impurities in the salt which can cause rapid initial corrosion and (b) changes in the alloys chemical composition, at least near the surface, due to non-equal leaching corrosion of alloy elements. Improved quantitative long-term corrosion rate predictions may be partially enabled by computational simulation methods and modern surrogate model methods, which have the potential to predict materials compatibility and performance in molten salts over long timescales.
Sparse experimental data alongside limited physicochemical simulations may pose a challenge towards achieving quantitative long-term predictions. It is not feasible to collect long-term data for all possible materials and all possible salts. To address this, new capabilities need to be developed, verified, and validated. The new prediction methods must consider the sensitivity of results to unknowns and uncertainties to provide acceptable safety margins under both normal operation as well as beyond design-basis accidents. A promising route for addressing the sparse data challenges is to use simulations that are enhanced through the use of artificial intelligence (AI), machine learning (ML), and/or data science (DS).
We will refer to this as AI/ML/DS augmented simulation. In some cases, an AI/ML/DS model can also make a direct prediction, in which case it is termed a surrogate model.
This report assesses the physicochemical modeling methodologies that are currently used to describe and predict molten salt corrosion of structural materials, including where challenges and opportunities lie. This report also assesses AI/ML/DS augmentation technologies presently being studied that have the capability to overcome data sparsity as well as enable more precise quantitative long-term predictions. The data used for training AI/ML models could potentially involve data from accelerated corrosion testing. The report focuses on illustrative, key examples rather than providing an exhaustive review of all work carried out to date.
As a demonstration of how AI/ML/DS can be used to augment simulations of relevance to molten salt reactors (MSRs), a modern ML surrogate model was applied to predicting the thermodynamic activity of elements in alloys. It was shown that the ML surrogate model was able to return accurate predictions and uncertainties on the order of 1000 times faster than a full chemical thermodynamic calculation.
Several other examples showing advances in applying AI/ML/DS to corrosion are also noted in the report, including one which predicts corrosion rates directly with typical predictions correct to within one order of magnitude. For each of these examples, significant quantities of training data are required for which the desired values are already known. Minimum sizes for adequate training sets may be as small as tens of data points or may be larger than tens of thousands of data points, depending on the method and the problem being addressed. Explainability/interpretability of AI/ML models is also important, and there have been advances and ongoing research in explainability/interpretability as described in Appendix 2.
10 The recent advances in AI/ML/DS have shown a demonstrated capability to fill data gaps with superior performance to any earlier technologies. Some of these technologies have also shown the capability to provide uncertainties, sensitivities, and identification of important physical factors/attributes. They have the potential to enable more precise quantitative long-term corrosion predictions even with sparse data, though will require coordinated or systematic experimental data collection for suitable development of the AI/ML models. Key observations of what is needed to realize the full potential of these methods are also provided within the report, and the report ends with a short summary.
11 Acronyms Used
_a Elemental Activity AI Artificial intelligence BPE Bayesian Parameter Estimation BSE Backscattered Electron CALPHAD Calculation of Phase Diagrams CPU Central Processing Unit DARPA Defense Advanced Research Projects Agency DS Data Science EDS Energy-Dispersive Xray Spectroscopy GBR Gradient Boosting Regression GIF Generation IV International Forum GP Gaussian Process GPU Graphics Processing Unit LASSO Least Absolute Shrinkage and Selection Operator MAE Mean Absolute Error MD Molecular Dynamics ML Machine learning MSR Molten Salt Reactor MSRE Molten Salt Reactor Experiment NN Neural Networks NPP Nuclear Power Plant ORNL Oak Ridge National Lab OLS Ordinary Least Squares PDE Partial Differential Equations p-GP Piecewse Gaussian Process PREN Pitting Resistance Equivalent Number RBF Radial Basis Function RF Random Forest SISSO Sure Independent Screening Sparsifying Operator SVM Support Vector Machine UQ Uncertainty Quantification
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- 1. Introduction After earlier interest in molten salt reactors (MSRs), there has been a resurgence of interest in MSRs, partly based on the Generation IV International Forum (GIF) inclusion of molten salt based reactors in the list of next generation of nuclear reactors.1 MSRs have the potential to play a role in addressing the huge challenges facing future energy production and thermal storage technologies. Within MSRs, molten salts may be employed as fuels, coolants, thermal storage media, with benefits from low vapor pressure, strong negative temperature coefficient of reactivity and avoidance of excessive heat generation by radioactive decay.2-4 Molten salts based on fluorides and chlorides are being considered across a broad spectrum of MSRs.5, 6 Liquid fuel MSRs have the unique characteristic of circulating molten fuel and fission products. It is desirable to acquire information on key physicochemical phenomena occurring in the molten salt liquid, containment vessel, and dissolved solid phases. These should be categorized according to their safety and risk importance for operating scenarios that might be encountered with either fast spectrum or thermal spectrum molten salt fueled and molten salt cooled reactor concepts currently being pursued by industry (such as Kairos Power, Terrapower, Terrestrial Energy, ThorCon, and Alpha Tech).7 Corrosion of structural materials in molten salts is a critical degradation phenomenon and is important in the technical realization of MSRs, as identified in the Oak Ridge National Lab (ORNL) Molten Salt Reactor Experiment (MSRE).8 Although the impact of molten salt corrosion on the structural integrity of materials has been studied for decades, and corrosion resistant materials have been identified, there are gaps in understanding of the governing mechanisms, and correspondingly large gaps in our ability to predict corrosion rates.9 Multiple corrosion mechanisms such as galvanic corrosion, anodic dissolution, impurity driven corrosion and thermal gradients can occur simultaneously, increasing the complexity in determining degradation kinetics. More quantitative information may be needed on the role of salt chemistry on surface reactions, anion-cation chemical interactions and anion/cation mass transport through the surface corrosion layers into the metal; and this lack of knowledge further exacerbates material selection and design protocols.10, 11 Enhanced abilities to predict quantitative long-term materials degradation, based on quantitative understanding of materials compatibilities with harsh salt and radiation environments, would be helpful.
Todays experiments, modeling, and monitoring at short timescales (minutes, days, and even months) are not fully capable of precisely quantitatively predicting long-term materials degradation (years). This is primarily because the corrosion rate changes as a function of time, and some of the terms governing long-term corrosion rates can differ from those of short-term corrosion rates, resulting in significant inaccuracies if long-term corrosion rates are simply extrapolated from short-term corrosion rates (Figure 1). The intrinsic corrosion rate is an instantaneous rate, which would be constant with perfect mixing at a given temperature if the surface area, surface alloy composition, and salt composition did not change over time. However, in practice, systems with metal alloy corrosion in molten salt are dynamically affected by various phenomena, including changes in surface area, mass transport limitations, and (importantly) alloy elements leaching into the salt. These dynamic system behaviors cause changes in which specific intrinsic rate is operative at a given time. The intrinsic rate can accelerate or decelerate due to these changes, leading to different long-term corrosion rates relative to short-term corrosion rates. In particular, there is evidence that the alloy and molten salt combinations currently being considered seriously for use in MSRs tend to have a fast initial corrosion rate followed by a slower long-term corrosion rate. It is presently believed that after initiation the primary factor for continued changes in the intrinsic corrosion rate are changes in the alloys chemical composition, at least near the surface, due to varied leaching corrosion rates for different alloy elements. The early time corrosion rate can also be affected by some types of impurities. As further noted later in this report, the slower long-term corrosion rate is thus
13 believed to be largely rate-controlled by slower long-term diffusion of elements out from the alloy (which we distinguish from slowing down due to passivation, as will be explained).
Figure 1. Illustration of the concept that long-term corrosion rates can differ from short-term corrosion rates.
This prediction gap may be partially addressed by computational simulation methods and physical insight informed modern surrogate model methods (which are each explained further below). These enhanced prediction methods have the potential to predict materials compatibility and performance for MSRs over operational timescales (years). A major challenge towards this goal is that the long-term predictions will be based on sparse data and limited physicochemical simulations. New domain-specific capabilities need to be developed, verified, and validated. Appropriate training data must be acquired, which could include data from accelerated corrosion testing. The new prediction methods must consider the sensitivity of results to unknowns and uncertainties to provide acceptable safety margins under both normal operation as well as beyond design-basis accidents. At the same time, overly conservative predictions could delay the application of practical technologies. Today, even the most advanced nuclear engineering simulation tools are tailored to conventional reactors with stationary solid fuel and lack the capability to model the chemistry and physics of liquid-fueled MSRs. As new simulation tools are developed, a promising route for addressing the sparse data challenges is to use simulations that are enhanced through the use of artificial intelligence (AI), machine learning (ML), and/or data science (DS).
We will refer to this as AI/ML/DS augmented simulation. In some cases, the AI/ML/DS model can serve as
14 a fast approximation for a portion of the calculations; when used in this way, the AI/ML/DS model is called a surrogate model.
This report aims to (a) highlight the physicochemical modeling methodologies that are currently used to describe and predict molten salt corrosion of structural materials, including where challenges and opportunities lie, (b) highlight AI/ML/DS augmentation technologies presently being studied that have the potential to overcome data sparsity as well as enable more precise long-term predictions. The criteria for corrosion-induced degradation of structural materials in MSRs being focused on in this report are the following:
- Loss of useful/sound metal due to corrosion attack (e.g., depth of attack)
- Critical depletion of strengthening elements (e.g., Cr as solid solution strengthening element)
- Corrosion-induced dissolution of strengthening phases (e.g., carbides)
This report assesses the physicochemical modeling methodologies that are currently used to describe and predict molten salt corrosion of structural materials, including where challenges and opportunities lie. This report also assesses AI/ML/DS augmentation technologies presently being studied that have the capability to overcome data sparsity as well as enable more precise long-term predictions.
The report focuses on illustrative, key examples rather than providing an exhaustive review of all work carried out to date.
One of the examples is a demonstration that is original to this report, showing how AI/ML/DS can be used to augment simulations of relevance to MSRs. In the demonstration, a modern ML surrogate model was applied to predict the thermodynamic activity of elements in alloys. It was shown that the ML surrogate model was able to return accurate predictions and uncertainties on the order of 1000 times faster than a full chemical thermodynamic calculation. Several other examples showing advances in applying AI/ML/DS to corrosion are also provided in the report, including one which predicts corrosion rates directly with typical predictions correct to within one order of magnitude. For all of these examples, significant quantities of training data are required, where the training data consists of answers that are already known. Minimum sizes for adequate training sets may be as small as tens of data points or may be larger than tens of thousands of data points, depending on the method and the problem being addressed. Explainability/interpretability of AI/ML models is also important, and there have been advances and ongoing research in explainability/interpretability as described in Appendix 2.
The recent advances in AI/ML/DS have shown a demonstrated capability to fill data gaps with superior performance to any earlier technologies. Some of these technologies have also shown the capability to provide uncertainties, sensitivities, and identification of important physical factors/attributes. They have the potential to enable more precise quantitative long-term corrosion predictions even with sparse data, though will require coordinated or systematic experimental data collection for suitable development of the AI/ML model. Key observations of what is needed to realize the full potential of these methods are also provided within the report.
- 2. Background and State-of-The-Art 2.1 AI/ML/DS Augmented Physicochemical Modeling AI, ML, and DS have made a significant impact in many technological fields in recent history, often in a supporting role. Various fields of relevance to MSRs (Chemistry/Materials Science, Chemical Engineering, Fluid Dynamics) have individually explored the use of AI/ML/DS for several applications:
gaining scientific insights from data, surrogate models, interpolation of data, extrapolation from data, and
15 achieving more efficient or more accurate simulations. Several trends and technologies have shown particularly strong performance, while others have shown signs of opportunities. AI, ML and DS are not wholly disconnected fields: they have overlap, and many applications of AI/ML/DS use all three. The use of AI/ML/DS can allow accurate predictions from sparse data, as well as improve computational calculations efficiency. To better inform the reader about the potential of AI, ML, and DS for overcoming challenges in MSR research, development, and safe deployment, we will review key aspects of these methods and provide recent examples that address the types of challenges encountered in MSR modeling.
Multiple definitions have been published for AI.12 For this report, the AI being discussed refers to machine-based systems (here, software programs) for which the outputs are changed based on exposure to an external stimulus or an initial dataset, toward a defined objective. This definition is consistent with the definition of AI used in the NRC Artificial Intelligence Strategic Plan.13 With this definition of AI, even simple linear regression models can be considered as AI. In this report, ML is defined as the subfield of AI that is focused on processes by which the program learns which output to create going forward as a function of future inputs (such as physical conditions).14 This definition is consistent with the definition of ML used in the NRC Artificial Intelligence Strategic Plan.13 AI and ML are complementary to (and in some cases reliant upon) DS, which is a field that has evolved from statistics and is related to the science of studying data to gain knowledge or insights.15 The NRC Artificial Intelligence Strategic Plan notes that DS involves classifying, predicting, and suggesting outcomes from data, which thus overlaps with AI. For the purposes of this report, DS will be used to refer to the use of data methods that require method-specific amounts of data in order to yield meaningful analyses: Some methods may require only a few datapoints while other methods may require millions of datapoints. In practice, today, different AI are often classified by the application or problem being solved, while ML is often classified by the type of methods used for the training of a particular AI. AI/ML/DS technologies are often able to fill gaps in data with high accuracy, and with uncertainty quantification available for some types of predictions. AI/ML/DS research has recently made significant progress in prediction accuracy through advances in ML, rather than through advances in AI and DS, and this report will correspondingly have greater emphasis on ML methods. For the purposes of this report, the definitions of AI/ML/DS are less critical than recognizing that new AI/ML/DS tools have the potential to solve practical problems and recognizing that these tools need to be evaluated. Further explanation of AI/ML/DS methods is provided in the Appendix, as a primer for readers less familiar with these methods.
One focus of this report is how AI/ML/DS can be used to overcome data sparsity to make predictions. There is typically no clear definition of when data is sparse versus dense, and the term sparse data has multiple meanings.16-19 However, it is recognized that data exists on a spectrum with "sparse" and "dense" on opposing ends of the spectrum, and case-bycase thresholds used for designating sparse or dense data. When the data is not overly sparse, predictions between points can be made using advanced interpolation methods.20-22 When data is very sparse, conventional interpolation methods will fail, and alternative methods based on surrogate models must be used. The surrogate models can be from one of several categories: partially physics-based models, phenomenological models (no physical basis, and uses a simple model to attempt to reproduce the behavior), complex surrogate models (no physical basis, and uses a complex model to try to reproduce the behavior). There is also the case of physics-informed models, which will be explained further below. In the ideal case, the model will adequately approximate the true behavior of the system that gives rise to the data, allowing for accurate predictions of values of unmeasured points. The main risk of having sparse data is the risk of overfitting,19 which results in accurate recalculation of the points used during training, but inaccurate predictions for the values between or beyond of the training points.
AI/ML/DS methods have shown tremendous ability to predict the existence of chemicals and materials not previously discovered,23-25 as well as the physical properties of known and even hypothetical materials.25-30 These advances have been largely spurred by the creation and population of large publicly
16 accessible databases containing chemical / material structures and properties. AI/ML/DS technologies can also be used for identifying (classifying) chemicals and materials from experimental characterization data.31, 32 In a completely different but equally important application, called symbolic regression, AI/ML/DS methods have been utilized for finding new expressions and relations (that is, equations) for predicting the properties of chemicals as based on physical constants (such as atomic radii). One notable example for finding such expressions is the Sure Independent Screening Sparsifying Operator (SISSO),
which will be described further below. Another notable method is the sparse identification of nonlinear dynamics (SINDy) algorithm.33-35 These symbolic regression models then provide physical insights based on which physical parameters are most important for a particular property. For some cases, reduced order models produced by symbolic regression with ML even converge to physically known equations, which is encouraging.33, 36 The inputs to the methods noted in this paragraph (including symbolic regression) typically involve large datasets of chemical/material atomic structures as well as measured (or simulated/calculated) properties. For many problems, a strength of ML is its ability to produce very accurate data, while a weakness is that the lack of physics knowledge results in a low ability to extrapolate beyond the training set accurately or to transfer predictions to new systems. To address these shortcomings, a growing area of research is physics-informed ML, which trades some accuracy for known points to gain physical realism, and thus allows for extrapolative predictions (as illustrated in Figure 2).37, 38 Currently, there is an untapped opportunity to apply ML methods to experimental datasets of transient data (like corrosion rates), but it can only be realized if the challenge of large datasets can be surmounted; this challenge should be surmountable for some experimental systems with current (and future) advances in robotics and automation (in part due to ML).39, 40 Accelerated corrosion testing experiments may also play a role in providing the data necessary for AI/ML/DS training and subsequent prediction.
In considering the usefulness and potential of AI/ML to make predictions for sparse data and computationally efficient surrogates for computationally expensive or unavailable models, it is useful to compare AI and ML to the historical background of finite difference and finite element methods which are used in the evaluation of differential equations. In the 1950s, finite difference and finite element methods were recently invented and being used to evaluate partial differential equations (PDEs), but often gave results which had very large errors during evaluation, where the numerical errors accelerated towards infinite error as the evaluation progressed. After sufficient research and usage, the numerical analysis community created more robust solvers and guidelines. Still, practitioners need to be a bit careful as modern methods can still have numerical errors which accelerate towards infinite error for certain problems. ML is in a similar early stage at present, as the performance of ML models is quite fragile and sensitive to the choice of the hyperparameters. At present, especially in the hands of non-experts, it is still quite common for ML to occasionally or sporadically give predictions with large errors, so some care is required. However, at present, as will be described below, AI/ML/DS can already be used for increasing efficiency of simulations, gaining insights, and filling data gaps. Trustworthiness of the outputs is important in safety-related applications. Currently, expert development is required when reliability is important, but the technologies for AI/ML continue to mature in their predictive accuracy and reliability.
An important field of research for trustworthy outputs is explainable/interpretable AI, and information about this topic is provided in Appendix 2.
17 Figure 2: Schematic illustration of accuracy and transferability of (a) traditional physics equations atomic potentials (b) mathematical ML atomic potentials and (c) physically-informed ML interatomic potentials. Accurate values for energy-volume relation for a particular structure, as obtained by density functional theory electronic structure calculations are shown with points (both filled points and unfilled points
), and the predictions of the three model potentials are shown with lines. A subset of the points was used for training the potentials: The points inside the training domain are shown by filled circles (
), and the points outside the training domain are shown with open circles (
). In the schematic, the two ML models are more accurate in the training domain (relative to simply using the physics-based equations). When looking at the full range shown, mathematical ML is the most accurate for interpolation, while physically-informed ML balances qualitative interpolation accuracy against extended extrapolation ability. Figures reproduced from reference 38.
2.2 Corrosion Prediction in Molten Salts When considering the prediction gaps, challenges, and recent developments for MSR physical modeling to describe and predict corrosion-induced degradation of structural materials, we first recognize that corrosion is inherently a multi-scale phenomenon. While more than two scales exist, it is convenient to divide the physical modeling into atomic-scale information (including both molecular-scale chemical kinetics and molecular-scale chemical thermodynamics, at < 100 nm) as well as continuum-scale information (including fluid dynamics, concentration gradients, heat transport, macroscopic materials properties, macroscopic chemical thermodynamics, and macroscopic chemical kinetics). Existing knowledge and understanding is insufficient for precise quantitative prediction of long-term (years-scale) material compatibility for arbitrary metal alloy and molten salt combinations.41 Long-term material compatibility can currently only be reliably ascertained through long-term testing. Very limited numbers of long-term experiments have been conducted, and it is not feasible to conduct long-term studies on all alloys and all salts. It may be possible to conduct a greater number of experiments using accelerated corrosion testing,42 though models that purport to predict long-term corrosion from accelerated corrosion testing results would need to be validated. For modeling of operating and potential accident conditions, a complete characterization of the accessible chemical space of molten salts and dissolved solutes is necessary (for systems under serious consideration).41 Accordingly, there is a need for predictive models of the phase diagrams for alkali halide salts and solvation thermodynamics of transition metal ions. This is a necessary and arguably urgent priority to understand corrosion mechanisms and design limitations.
Additionally, it may be necessary to acquire information on the solvation thermodynamics of heavy metals, lanthanides, and actinides and more importantly to compute the thermodynamic stability of all potential solid phases.
Several efforts have constructed thermodynamic databases capable of predicting density, heat capacity, thermal conductivity, and free energies of molten salt mixtures.43, 44 However, these efforts are based on empirical fitting equations and are prone to some known systematic errors. The experimental difficulties in measuring trace concentrations (and a reliance on extrapolation to outside the measured ranges) have led to some compositions, pressures, and temperatures for which reliable data is limited.
18 Solubilities of transition metal ions and their complexes (metal halides, hydroxides, and oxides) are particularly difficult to cover experimentally because of their large range and variety.
In corrosion science, passivation typically refers to the formation or addition of a stable layer at the surface which acts a shield against corrosion. Several data collection efforts have focused specifically on attempts to identify stable passivating metal oxide surface layers in the presence of molten salts.45 The authors of this report have not seen any information indicating that passivating layers that are stable at relevant temperatures have been achieved for any molten salt and metal alloy combination pairing under serious consideration for nuclear MSRs (Figure 3). Much of the research for high temperature molten salt corrosion is related to non-nuclear applications, such as for concentrated solar power applications.46 Shared knowledge between the fields, through scientific literature or other mechanisms, may be beneficial for advances in high temperature molten salt corrosion mitigation and prediction. Within the broader context of high temperature molten salt corrosion studies (not for nuclear reactors), there is some evidence that TiO2 may be a stable layer47 in CaCl2-CaO if formed, that a TaCl layer has the potential to be formed as stable layer47 from Ta containing alloys, that NiO may form a stable layer45 in LiCO3, and that and NaFeO2 may form as a stable layer45 in NaNO3. However, to our knowledge, no system has been found that would have a layer that is both stable and sufficiently protective to be called a passivation layer under MSR conditions, within the context of a nuclear reactor. Even when oxide or other chalcogenide layers are formed during/before an alloy comes into contact with a molten salt, the layer typically becomes dissolved48 once in contact with a molten salt, and/or does not prevent leaching of alloy elements from behind the layer (see discussion in refs 49, 50). Another approach that can mitigate corrosion is boronization.51 While micrometer scale pitting has been observed as a corrosion mechanism in molten salts for some alloys,51 pitting at the mm scale has not (to our knowledge), been observed for typical molten salt corrosion measurements with alloys that are under consideration for use at this time. Pitting is typically associated in the larger field of metals corrosion with systems that have passivation. Thus, if successful passivation is later successfully developed for MSRs, then long-term pitting predictions would become one of the terms in precise quantitative long-term predictions.
Figure 3. Illustration depicting how, under high temperature conditions, passivating layers are known to form or be achievable for some metals with that can act reduce corrosion rates in high temperature air or steam environments, but such passivating layers are not passivating in relevant high temperature molten salt environments. Figure adapted from one provided by Wendy Reed of U.S. Nuclear Regulatory Commission.
A few studies have dealt with models to predict corrosion rates of structural materials during exposure in molten halide salts, typically in the absence of radiation.52-56 For the main contributions to
19 long-term corrosion, the body of research on molten salt corrosion of alloys has ultimately found that mass transport of elements in the metal solid to the surface is important, with grain boundaries and local elemental concentrations in the alloy being important factors, as described in references cited throughout this report. In this report, we do not consider the effects of radiation nor of the chemicals that are produced in these systems as products of radioactive decay. However, one study has noted that tritium fluoride can form under neutron radiation when Li and F are present in the salt and the tritium fluoride can then act as an oxidant.50 Impurities in molten salts can also drive high initial corrosion rates, such that molten salt purity is an important technical consideration, but outside of the focus of this report.8, 42, 57-60 It is worth noting that preparation for the ORNL MSRE experiment had involved extra efforts for salt purification, but that the referenced report does not provide the details of the salt purification steps.
Moisture from graphite could potentially also add as a short-term oxidant source in MSR designs that have direct contact between graphite and the salt, as well as chemical reactions involving graphite and metal structural materials.61 The aforementioned contributors to corrosion from factors other than the molten salt are not the focus of this report: this report focuses on the long-term corrosion by the contact of the metal with the molten salt, in the absence of other contributors.
One of the advances in modeling occurred by including the effects of the elemental diffusion in the alloys bulk. DeVan and Evans52 calculated corrosion rates driven by Cr depletion in the Nibased alloy INOR8 (developed for use in MSRs) during exposure in pre-equilibrated NaF-ZrF4UF4 at 800°C for 18 months. The model used an error function solution for diffusion in a system with constant surface boundary condition (constant Cr concentration given by equilibrium reactions with the fluoride salts) and relied upon a self-diffusion coefficient of Cr measured in INOR8. McCoy et al.53 correlated the role of Cr diffusion in the alloy to observed corrosion rates of Hastelloy N specimens inserted in NaBF4NaF (92-8 mole%) thermal convection loops with 607°C and 460°C as the temperatures in the hottest and coldest sections, respectively. There were differences between measured Cr loss and the models predicted Cr loss, which were attributed by the authors to the role of faster grain boundary transport and the presence of impurities. Cho et al.54 studied the impact of thermal gradients and fluid flow induced by natural convection gradients on the corrosion rate for the selective oxidation of chromium in alloy 230 for exposures in a eutectic KCl-MgCl2 salt mixture between 600°C and 950°C. Dimensionless parameters were utilized to describe heat and mass transfer in the system and predict average corrosion rates, although it only considered the effect of mass transfer and did not consider the local electrochemical environment.
Mehrabadi et al.55 extended the approach presented by Cho et al. to include the electrochemical transfer between the alloy 230 specimen, molten salt (eutectic KCl-MgCl2 mixture) and containment material (Ni) in a steady state 3D-model. Corrosion potentials and corrosion rates were calculated by considering electrochemistry of the most relevant reactions. The electrochemical surface corrosion kinetics were coupled with heat and mass transfer while simultaneously considering the effect of the critical corrosion parameters on the corrosion rate (e.g., temperature, grain boundaries properties). The corrosion rates obtained by the model in terms of the current density and average corrosion potential were in good agreement with experimental results with less than a 7 % difference. However, no images or micrographs of the corroded specimens were presented and the chosen exposure time of 5 h for the thermosiphon experiments54 seems extremely low for validation of a model aiming to predict corrosion rates since the thermodynamic driving forces for corrosion will be expected to be the highest in the initial stages. Zheng et al.56 modeled the Cr depletion from 316 stainless steel exposed in molten FLiBe salt at 700°C for up to 3000 h. An error function solution of Ficks second law using an effective diffusion coefficient for Cr was employed to calculate the time evolution of Cr depletion in the steel. The effective diffusion coefficient was estimated from mass change measurements assuming the measured mass change corresponded to Cr depletion in the alloy. A good agreement between measured and calculated Cr concentration profiles in 316 stainless steel was achieved for the exposure in FLiBe salt for 3000 h at 700°C. The spatial information of grain boundaries and the locations of impurities cannot typically be known for corrosion
20 experiments and applications; therefore, there will always be some uncertainty in the predictions. The possibility of attainment of steady state corrosion rates or influence of other changes in the system on the long-term corrosion rates cannot be ascertained with such a model. These approaches are limited to surface reactions and do not consider chemical interactions of alloying elements in a multicomponent alloy system and thereby ignore altered driving forces for species transport during corrosion on the surface. Ultimately, the assumption of diffusion being driven by solely depth-averaged concentration gradients in these models can either lead to significant deviations from reality or limit the capability of these models to predict simultaneous depletion-enrichment processes. Diffusion is governed by gradients in chemical potentials, which depends on the local environment (not just macroscopic averages), and this is important to consider in a multicomponent and multiphase alloy.
The aforementioned models which include diffusion in the bulk are useful tools to predict surface corrosion rates but do not consider corrosion-driven chemical interactions between alloying elements nor other factors which affect the transport processes within the alloy. In these systems, long-term corrosion rates (over years) are substantially different from short-term corrosion rates (such as over tens of hours).62, 63 The aforementioned simple diffusion models overestimate long-term corrosion since they have been formulated to predict initial corrosion rates and the thermodynamic driving forces for corrosion will be expected to be the highest in the initial stages. As elements are leached out of the alloy, diffusion within the alloy will also occur at these temperatures, and the intrinsic corrosion rates defined by the surface and near-surface will be affected by the elemental diffusion within the metal alloy. The activities of the individual elements, as a function of time, need to be included in corrosion calculations for greater accuracy relative to only using their concentrations. It is important to note that there are two different reasons that initial corrosion rates are higher than long-term corrosion rates, for these systems: (1) impurities can cause high initial corrosion rates (as noted earlier), (2) the intrinsic corrosion rate decreases substantially as the near-surface region gets depleted, after which the effects of mass transport (diffusion of elements in the solid) become more important. For example, in the work by Pillai et al.62 with KCL-MgCl2 salt and alloys containing Fe, Ni, and Cr, it was observed that the corrosion rate was more rapid at ~500 h but had stabilized before ~1600 hours. To address the shortcomings from looking at only initial concentrations or neglecting diffusion within the alloy, a thermokinetic modeling approach64 was employed to quantify corrosion rates and predict microstructural evolution of metallic materials during exposures in molten chloride salts.65 This approach models simultaneously occurring corrosion, diffusion, and dissolution processes in the alloys by considering all relevant chemical interactions between the alloying elements. Composition and temperature dependent chemical potentials and mobilities are extracted from calculation of phase diagram (CALPHAD) based thermodynamic and kinetic databases. The one-dimensional coupled thermodynamic-kinetic model has shown to be capable of successfully predicting the corrosion-induced compositional and phase changes in multicomponent and multiphase alloys.64, 66-70 of the desired simulation time. This is a key advantage to overcome the limitations of domain size, number of alloying elements and computational time imposed by lower scale modeling methods.
The model provides spatiotemporal evolution of changes in compositions, chemical potentials, and element mobilities for different surface conditions. And element mobilities for different surface conditions.
Pillai et al.62 performed computations for the case of a binary NiCr and multicomponent Nibase alloys 230, C276 and 600 exposed to binary commercial KCl-MgCl2 salt mixtures at 700. The modeling approach discussed earlier was able to predict the molten salt-induced compositional changes and phase transformations in all the studied alloys. For example, Figure 4a shows the backscattered electron (BSE) image as well as energy-dispersive xray spectroscopy (EDS) spatial elemental maps (Cr, Mn, Mo, W, Fe and Ni) for C276 after 1000 h in purified commercial salt at 700 °C. The commercially available salt contained 17.4%K, 18%Mg, 2.3%Na, 62.2%Cl (mol.%) and impurities including 5% H2O and lesser amounts of Br, Fe, B, Mn and SO4. The comparison between measured and calculated Cr, Mo, and W concentration
21 profiles in Figure 4b shows that the model has good agreement with the experiment and reproduces the simultaneous depletion of Cr in the alloy subsurface concomitant with enrichment of Mo and W at the alloy surface. These concentration changes were suggested to result in the potential formation of brittle intermetallic phases. The experimental depletion is not homogeneous (Figure 4a) and the averaged concentrations, as a function of depth (Figure 4b and Figure 4c), show kinks. These features will be commented on further below. The authors suggested that the inert nature of W and Mo in molten halide salts is expected to retard corrosion attack under the test conditions. Notable conclusions from the work were: 1) It is essential to evaluate how the capsule material, specimen material, and salt composition affect the intrinsic corrosion processes through electrochemical transfer reactions. 2) When predicting corrosion behavior in molten halide salts, the chemical activity of Cr and its diffusion in the alloy were found to be more relevant than the Cr atomic percent in the alloy. The chemical activity of Cr depends on the local concentrations of other elements present, and the crystal phase of the solid.
(a)
(b)
I Figure 4: (a) Backscattered electron image and corresponding EDS elemental maps (Cr, Mn, Mo, W, Fe and Ni) for C276 after exposure for 1000 h in purified commercial salt (commercially available salt containing 17.4%K, 18%Mg, 2.3%Na, 62.2%Cl (mol.%) and impurities including 5% H2O and lesser amounts of Br, Fe, B, Mn, and SO4) at 700 C; corresponding comparison between measured (symbols) and calculated (lines) (b) Cr and (c) Mo and W concentration profiles.65
22 In another work by Pillai et al.71, the authors successfully verified the hypothesis65 about Cr activity being one of the key parameters in predicting alloy corrosion rates in molten salts.71 The Cr activity is modulated by the overall elemental composition of the alloy, as well as the temperature. To assess the influence of these additional factors (composition and temperature), and to ensure that the activity remained as a dominant term, additional experiments were performed. Model binary, ternary, and quaternary Febased and multicomponent commercial Nibased alloys were exposed in the binary eutectic KCl-MgCl2 salt at 700 °C and 800 °C in quartz capsules for 100 h. The dissolution rates of Cr from the alloys were estimated from previously measured72 corrosion rates of pure Cr in the same salt. Cr depletion from the alloys was calculated with the coupled thermodynamic-kinetic model64 and compared with measured depletion profiles in the alloys and correlated to the measured Cr content of the salts after exposure. The Cr activity informed predictions for microstructural evolution (element distribution and compositional changes) in the alloys during corrosion were in very good agreement with experimental observations, suggesting an improved physical agreement between the model and the underlying mechanisms. Figure 5 compares the measured Cr content in the salts with the predicted Cr loss after exposure for 100 h in the binary KCl-MgCl2 eutectic (68:32 mol.%) salt mixture at 700 °C (Figure 5a) and 800 °C (Figure 5b). The results demonstrate that the model was well able to predict the time with varying temperature and alloy composition for dissolution of Cr in the binary KCl-MgCl2 salt. Predominant dissolution of Cr was confirmed with the measurements of the salt chemistry after exposure with the highest contents of Cr measured compared to other alloying constituents. Based on the results, it was confirmed that an alloy with a higher initial Cr activity was most likely to corrode faster for the first 100 h under similar conditions (temperature, salt chemistry, capsule material) but diffusion kinetics of the dissolving alloying constituent in the alloy will govern the long-term corrosion behavior. This is consistent with literature knowledge about the relative ordering of halide stabilities (Cr > Fe > Ni > Mo) and Cr diffusion (which is known to also occur along grain boundaries).73 Completely accurate prediction of long-term corrosion rate behavior would also require knowing the locations and extents of impurities and grain boundaries. Lack of complete knowledge about impurities and grain boundaries means there are parameters that cannot be mapped or quantified exactly. The effects of impurities and grain boundaries will thus, in practice, present as stochastic components of the dissolution rate. While this presents a challenge, there is an opportunity to use models which incorporate stochastic behavior during the dissolution to account for such inhomogeneities, as will be discussed further below.
23 (a)
(b)
Figure 5: Measured Cr content (inductively coupled plasma atomic emission spectrometry, ICP-AES) in the salt after exposure for 100 h in the binary KCl-MgCl2 eutectic (68:32 mol.%) salt mixture at (a) 700 °C and (b) 800 °C compared with the calculated Cr loss from the alloys under the same conditions 71 Phase-field models have also contributed to the success of short-term predictions and present a step-forward to accounting for grain boundaries. In a recent work by Bhave et al.74, the authors demonstrated the applicability of an electrochemical phase-field model to describe the corrosion of two binary NiCr alloys (Ni5Cr and Ni20Cr wt.%) in molten FliBe (LiBeFe2) at 700 °C for 1000 h. Multi-dimensional (1D, 2D and 3D) simulations were undertaken and validated against experimental data. The agreement between measured and calculated Cr depletions were deemed satisfactory in the case of the Ni20Cr alloy when the 1D simulations incorporated effective diffusion coefficients accounting for diffusion along grain boundaries. However, the same 1D model overpredicted Cr depletion in the Ni5Cr alloy. 2D simulations using reconstructed experimental microstructures underestimated the Cr depletions for both alloys while 3D simulations accurately predicted the Cr loss for the Ni5Cr alloy. The authors highlighted the computational costs associated with 3D phase-field simulations and it is worth mentioning here that the systems under consideration were binary alloys. Any extension of this approach to multicomponent alloys is currently unfeasible from a practical standpoint of employing the high-dimensionality models for lifetime assessment of MSRs over a typical operational duration of years.
Methodologies which can circumvent, reduce, or overcome the computational costs constraint will present opportunities for progress. Phase diagrams of multicomponent salt-metal systems are relatively more computationally intensive but can be obtained from a variety of reliable free energy calculation methods.75, 76 Current databases for molten salt thermodynamics use a quasi-chemical model44 adopted from the CALPHAD method. These models are structurally motivated, with fitting parameters for pairs and triples of nearby atoms. Phase diagram comparisons (and parameterizations) of CALPHAD analytical models across binary, ternary, and quaternary systems are a key component of design and validation of current models for MSRs.43, 77, 78 Although calculation methods appear to be well established, there is a lack of a systematic connection of computational results to analytical thermodynamic models, primarily due to a lack of dedicated funding programs to support the required efforts. Sufficient comparison between modeling results and experimental observations for thermodynamics is still lacking in the literature. The focus has primarily been on ion forcefield models and methods for computing ion free energies,79 diffusion constants,84 conductivities,84 and coordination numbers80-83, that have been compared to values extracted from analysis of experimental observables.
Accelerated corrosion testing is not yet used for long-term prediction of corrosion in molten salts but may also be an avenue for progress that would be synergistic with AI/ML/DS. In non-molten salt
24 corrosion testing (including for nuclear applications), accelerated corrosion testing is sometimes used to assess materials compatibility, with measurements shorter in time than the expected duty life.85-87 ASTM G31-21 notes that accelerated corrosion tests give indicative results only, or may even be entirely misleading.88 Four known ways to accelerate corrosion in molten salts are: (a) varying the temperature,89, 90 (b) the use of additives (such as oxidizing salts, H2O, graphite, or specific capsule materials as described further below),42, 58-60, 91-93 (c) increased flow rates94 (static tests are less predictive of corrosion under flow, than flowing tests), and (d) electrochemical/galvanic coupling95. Each of these methods have limitations in their ability to be mapped to non-accelerated corrosion, while still being useful. For example, galvanic coupling does not capture all of the corrosion. Studies to assess the importance of mass transport in the corrosion of molten salt, relative to galvanic controlled corrosion, found that mass transport and chemical activity gradients are important, and not captured by considering only galvanic controlled corrosion.73 Due to the importance of mass transport and chemical activity gradients, capsule materials which can incorporate leached elements can actually accelerate corrosion.93 While the choice of capsule material accelerating corrosion is normally regarded as undesirable, it could be very useful for intentional acceleration studies. A study looking at flow found that flow did not accelerate corrosion inside FLiNak.96 It is possible that the lack of corrosion acceleration from flow in that study is due to mass transport of the elements in the metal being rate-limiting under some conditions, such that their removal within the fluid is not strongly rate controlling. Systematic studies to understand how capsule materials, flow, and other controllable experimental choices can be used to accelerate mass transport in a way that can be mapped to the corrosion that would occur in an MSR under normal operating conditions. Even if corrosion acceleration methods are found to be capable of solely accelerating the initial portion of corrosion, that may still facilitate gathering experimental data for long-term corrosion rates. For example, one can imagine a testing method in which galvanic potentials or an excess of oxidative salts are used to create an initially corroded surface in several hours (to facilitate the initial corrosion that occurs in the first few thousand hours of corrosion), followed by changing to fresh salt to test the rate that would occur under long-term corrosion. There is evidence that such an approach might be viable.95 We are not aware of any studies which have taken such an approach for predicting long-term corrosion, so this approach remains to be investigated. Importantly, with the use of AI/ML/DS, it may be possible to map experimental accelerated corrosion testing data to long-term corrosion predictions, provided that sufficient characterization is performed. Mapping of accelerated corrosion testing to long-term predictions using AI/ML/DS was also considered for non-molten salt corrosion in nuclear waste applications at least as early as 1994.87 Methods to use AI/ML/DS to map accelerated corrosion testing results to long-term corrosion testing would need to be validated.
- 3. Current Modeling Challenges and Limitations A challenge that exists is that MSR degradation is not solely a chemical thermodynamics phenomenon and is also a chemical kinetics phenomenon. Dissolution requires kinetic models that couple fluid flows with surface chemistry, and leaching can also require considering mass transport from within the solid.97-100 As noted in the earlier sections, the chemical kinetics are not completely understood for metal corrosions in these environments: currently, quantitatively accurate models do not exist (particularly for long timescales). For quantitative operative first elementary step kinetic models, it would be necessary to know the chemical reaction network and the elementary step energy barriers within the operative local environment (particularly for surface reactions). It is generally not feasible to measure nor indirectly obtain such information except for specialized experiments that cannot be extended to chemically complex reaction environments. Presently, this mechanistic gap in knowledge is rapidly being addressed at two chemical scales: (a) with automatic reaction network generators which construct
25 reaction networks at the molecular or bond-breaking level,101, 102 and (b) with automatic elementary step reaction search tools,101-104 which explore potential energy surfaces to find elementary step transition state barriers. Research is presently being conducted to not only construct such tools, but to enhance them using modern ML paradigms, to accurately predict the reaction rates and operative pathways at the atomic/molecular level. In aggregate, reaction network and elementary step discovery tools will play a critical role in addressing the mechanistic gaps for first-principles multi-scale modeling.
Electrochemical models to describe molten halide salts induced corrosion assume the nature of underlying oxidation and reduction reactions occurring between the molten salt and the multicomponent alloy. Typically, oxidation of Cr is assumed to occur with formation of Crhalides,54 within the widely acknowledged and experimentally observed preferential dissolution of Cr from high temperature alloys.105 However, concurrent dissolution of other alloying elements such as Mn, Fe, Al, and Ti has been experimentally observed in molten chloride65, 71 and fluoride106 salts. Furthermore, the kinetics of the anodic oxidation and cathodic reduction reactions assume the reference chemistry of the alloy to be the bulk chemistry, but the corrosion processes can alter the subsurface alloy chemistry (tens to hundreds of microns). In the aforementioned cases, the driving forces for transport in the alloy were not considered, which consequently means that compositional changes of the dissolving element in the corrosion affected zone were not considered. Additionally, the chemical interactions between alloying constituents in a multicomponent system are typically ignored in these models, which can be significant, as was shown the case study shown in Figure 4, where Cr depletion was accompanied by enrichment of Mo and W in the alloy subsurface.
Models aiming to predict the compositional changes in the alloy require information on the dissolution rates of material constituents from the alloys during high temperature exposures in molten halide salts. This data is primarily derived from experiments providing quantitative information on the corrosion rates of the studied alloys (computational studies are discussed further below). The data is typically collected by measuring mass change56 or elemental depletion in the alloy62 during high temperature exposures in the molten halide salts. However, the lack of a standard procedure for testing corrosion in molten salts results in non-representative measurements which complicates quantifying the extent of corrosion damage. The main variables resulting in discrepancies of reported corrosion rates are a lack of consistency in salt chemistry (different purification procedures), use of different container materials (dissimilar capsule/test material combinations) and employing distinct test procedures (capsule/loop tests). Furthermore, the role of oxygen and hydrogen related chemical impurities (such as O2, H2O, OH-, and H+) in governing the corrosion kinetics is not well understood. Oxygen and hydrogen related chemical impurities accelerate corrosion by serving as oxidants with a high enough redox potential to readily accept the electrons released during anodic dissolution of metallic elements during the corrosion of metals and alloys. To properly understand the role played by oxidants, the cathodic current densities associated with the reduction reactions need to be quantified. Additional non-electrochemical factors, such as the alteration of the surface by the impurities, the effects of microstructural heterogeneity on corrosion, and the possibility for fluxing mechanisms in which scales form and dissolve repeatedly may also play an important role in accelerating corrosion.
For physicochemical based chemical kinetic modeling, the thermodynamic equilibria still need to be calculated, and this is one of the computational bottlenecks. A significant fraction of the computational time and effort (~95 %) is spent on calculating the thermodynamic equilibrium at each node of the computational mesh in every time step. The time and effort needed for these calculations increases manyfold with the number of considered material constituents and phases. This problem is well known and has been reported before as the limiting factor in coupled thermodynamic and kinetic modeling approaches.107-109 Acceleration of the equilibria calculations is critical to facilitate rapid and efficient simulations of realistic component operational lifetimes between 30-50 kh. One method is through the generation of a pre-determined table of compositional space and conditions required for the kinetic
26 simulations which can then be used to create a lookup table to be used during multi-scale modeling. The dataset can be created with high-throughput thermodynamic calculations109 with input features such as temperature and chemical compositions. A ML surrogate model can then be trained on this dataset and provide the required information such as chemical potentials, constituent activities, stable phases and element mobilities during the simulations without resorting to computationally intensive equilibria calculations. A demonstration of this type of ML surrogate model is provided in Section 4.1. In conjunction with this strategy, higher dimensionality models (2D, 3D) utilizing phase-field methods110 can be used to resolve the microstructural features such as grain boundaries, but these too are computationally expensive and a realistic implementation of these models for lifetime predictions of multicomponent alloy systems is currently impractical.
To computationally predict the dissolution rates in a multicomponent alloy system, a quantitative knowledge of the relative free energies corresponding to the states of the system is necessary. There is currently a lack of computational simulations that fully consider the interactions of the multicomponent alloy surface with the molten halide salts during simulation of the time evolution of the material constituents and salt components. There is a lack of thermodynamic and kinetic data for the multicomponent metal-salts systems. Initial attempts to address this problem are currently being undertaken.78 This is further complicated by the fact that quantifying salt material interactions at atomic/molecular-scale under corrosion conditions is challenging (it is generally not possible to directly probe such interactions under actual conditions at the atomic/molecular-scale). An alternative approach to partially overcome this challenge is to quantify the time and temperature dependent dissolution of pure elements (e.g., Cr, Fe, Ni, Mn) in the technically relevant salts and integrate that information in a coupled thermodynamic-kinetic approach to predict the corrosion-induced microstructural evolution.
Such an approach was proposed by Pillai et al.62 as was discussed in Section 2.2.
ML can help with the coupled thermodynamic-kinetic approach. Consider, for example, the thermodynamics of molten salts. Complete analytical, but approximate, theoretical models for molten salt thermodynamics have been available since 1977.111 These included free energies and radial distribution functions for mixtures of charged hard spheres that formed the basis for reproducing structure factors measured by neutron diffraction. Additional work showed that the screening length from the theory gives a theoretical basis for the stability of complex ions dissolved in molten salts and capacitance at electrode interfaces.112 The same analytical model within a dielectric background113 was also used to predict aqueous solution viscosities, conductivities, and mutual diffusion. Despite its wide range of applicability (the only free parameters are ionic radii), the relative stability of solid, solute and liquid phases requires great accuracy to compute. Thus, its predictions need to be compared to molecular simulations and experiment, then carefully corrected to arrive at high accuracy predictions of densities, free energies, half-cell potentials, solubilities, and phase equilibria. Given that an efficient physics-based model already exists, any gaps in accuracy can likely be filled with error correction ML, such as neural networks (NN).
- 4. Opportunities for New ML Tools to Augment MSR Long-Term Corrosion Rate Predictions ML Augmentation of modeling can be roughly divided into two scales: atomic-scale predictions, and continuum-scale predictions. For atomic-scale chemicals thermodynamic properties predictions, AI/ML/DS is in a thriving status. There is a steady addition of new studies which are using computationally and/or experimentally obtained databases to create ML predictions of atomic-scale and materials-scale chemical predictions with unprecedented accuracy and at low computational expense.101, 114-117 The
27 challenge is now largely (but not completely) a balance between cost and accuracy. The methods used for atomic-scale predictions are generally supervised ML methods,101,114-117 where many labeled features/parameters are used (such as atomic number, atomic coordination numbers, atomic distances) to predict properties of interest (such as formation energy). A challenge in these predictions is that the most accurate extrapolative prediction methods are not directly extensible to complex materials (such as alloys with more than five elements). The "curse of dimensionality" inhibits accurate extrapolative prediction for composition spaces beyond two or three elements (for example, even a two dimensional material's cluster expansion becomes effectively computationally intractable for density functional theory electronic structure calculations once several species are included).118 When interpolative predictions do not require a model with physical realism, mathematical ML can be used to achieve high accuracy predictions if the training dataset is sufficiently dense.38, 119 However, it is not feasible to explore (even in a sparse way) the vast parameter space that complex material compositions span (such as with modern corrosion resistant alloys). Accelerated corrosion testing could be used to supplement the limited experimental long-term corrosion measurement data, but accelerated corrosion testing is likely not a complete solution, by itself. A promising route forward is to use © mathematical ML models when interpolation is sufficient while using physics-informed ML (or ML augmented simulations) for extrapolation and for sparse datasets. This leads to two sub-routes forward: (a) to use methods that work sufficiently well on sparse data21, 120-123 and (b) to use bottom-up approaches124 such that the larger scales can be calculated accurately from the smaller spatial-scales. AI/ML can play a role at the smaller size-scale as more accurate interatomic interactions can be produced using machine learning101, 125-127 to enable ML augmented first-principles calculations. These ML augmented first-principles calculations can then enable more accurate and more efficient atomic-scale simulations of corrosion processes.128, 129 Many examples exist for ML of physicochemical properties of materials, including atomic-scale chemical kinetics in alloys (related to embrittlement).130 We will include one demonstration and some literature examples of relevance to MSR corrosion.
For continuum-scale predictions, there are advances in the ability of ML augmentation and surrogate models to calculate time dependent behavior more efficiently or more accurately in computational fluid dynamics as well as heat transfer. The continuum-scale advances are primarily focused on general tool development, and thus will not require any dedicated efforts from the MSR research community: simply adoption of state-ofthe-art continuum-scale simulation codes as they become available. Accordingly, the subsection provided on ML augmentation and surrogate models for continuum-scale calculations is shorter than the other subsections, in the subsections below.
4.1 ML Surrogates for Predicting Metal Alloys Elemental Activities (Demonstration)
As discussed in the previous sections, AI/ML driven approaches can complement the computational methods at each scale by either reducing the computational effort (e.g., by estimating the thermodynamic properties without the need to perform expensive equilibrium calculations) or by synthesizing new data and configurations, estimating uncertainties and the inferring unknown properties of new salt-metal systems. This can result in AIdriven surrogate models which can significantly reduce the computational ef©ort in describing the complex corrosion-induced degradation processes while simultaneously retaining the precision of the high-fidelity physics-based models. In the following section, a case study is chosen to demonstrate the development of AI/ML surrogate models for elemental activity in alloys, to complement the 1D-coupled thermodynamic-kinetic model in describing the Cr depletion from Febased alloys during high temperature exposures in a purified chloride salt. The need for predicting elemental activities of alloys was described in an earlier section and noted that if accurate thermodynamic activity predictions can be made with an ML surrogate that can return predictions rapidly (relative to full
28 thermodynamic calculations), that such capabilities will enable multi-scale modeling of the corrosion of these alloys to become more computationally tractable.
The challenges described above must be solved by connecting together physics-based engineering models of MSR operating conditions, analytical models of solution thermodynamics (such as the mean spherical approximation and calculation of phase digrams), molecular dynamics computations of free energies, and quantum-mechanically based forcefield parameterization methods. Operational models of MSRs will determine the temperatures and material compositions of interest to corrosion modeling in solid-fueled and liquid-fueled MSRs. Analytical models can be routinely applied to predict key information on operational conditions: molten salt phase diagrams, conductivities, impurity solubilities at low and maximum concentrations, relative stabilities of solid phases, and electrochemical half-cell potentials.
Their adjustable parameters are physically meaningful values and need to be evaluated against experimentally obtained data as well as used to systematically correct the quantitative predictions of molecular simulations. These points of comparison must be collected and documented in machine-readable form. The resulting datasets can be used both for software and experimental quality assurance and as a means for rapidly assessing the reliability of new computational results.
As a demonstration of how AI/ML/DS can be used to augment simulations of relevance to MSRs, a modern ML surrogate model was applied to predicting the thermodynamic activity of elements in alloys.
For this example, the ML surrogate models ability to return accurate predictions and uncertainties was investigated. For individual elements in a metal alloy, the elemental activities do not depend only on their own concentration (as shown in Appendix 3); each elements activity also depends on the concentrations of the other elements in the alloy as well as the local crystal phase. The performance of a multi-linear regression model will also be compared to the ML prediction, to demonstrate the utility of modern ML.
As noted earlier, these thermodynamic activities can then be used in multi-scale modeling for corrosion rates simulations. The temperature range investigated is 600 to 800 °C, and the alloys comprise 10 elements (Fe, Cr, Mn, Si, C, Ti, Mo, Al, Nb, Ni), with concentrations chosen from the realistic range for Febased alloys. The training and testing data creation and validation were both based on thermodynamic calculations using values from the Thermo-Calc software for calculation of phase diagrams (CALPHAD) and thermodynamic databases TCFE11 and TCNI11.131-133 The element concentrations, the phases, and the temperatures mean that this is a prediction problem with a 13-dimensional input. A successful ML surrogate with reliable uncertainty for its predictions could enable ML augmented multi-scale modeling, if more rapid in its predictions than performing full CALPHAD calculations. Additionally, as noted, the ML surrogate could be built upon using experimental data and Bayesian methods, thereby utilizing both forms of sparse data to create useful predictions with acceptable uncertainties. The ML surrogate model used here is a piecewise Gaussian process based (pGP) surrogate model, and the technical details are included in Appendix 3. Figure 6 shows an illustrative comparison of the pGP surrogate model predicted activity and the CALPHAD calculated activity to visually demonstrate the low error of the predictions. In addition to the pGP surrogate model, a multi-linear regression ordinary least squares (OLS) model was fitted to the same initial dataset for a performance comparison.
29 Figure 6: Comparison of the modern AI/ML pGP surrogate model predicted activity and actual CALPHAD calculated activity for Cr, for 300 compositions with Cr concentration between 14 and 17 Atomic %. The Cr activity does not depend only on the concentration of Cr, and instead depends also on the alloy concentrations of the other elements, the temperature, and the phase (a 13-dimensional problem). There is good agreement between the p-GP model predictions and the actual calculated activities.
Following training and testing using 150,000 points, the activities of 15,000 points of alloy composition (generated within the same range) were used for validation and for demonstrating that the pGP surrogate model will outperform an OLS model. Table 1 provides a normalized error, based on taking the mean absolute error (MAE) from the validation points consolidated across each elemental activity and then dividing by each elements central composition (provided in Appendix 3). The pGP MAE is on the order of 1/10th that of the OLS MAE. The average normalized errors of the p-GP surrogate and OLS surrogate predictions are also show in graphical form in Figure 7. There is not good agreement between the OLS predicted activities and the target activities (calculated from CALPHAD), as demonstrated by the parity plots in Figure 8. In contrast, there is good agreement between the pGP predicted activities and the target activities (calculated from CALPHAD), as demonstrated by the parity plots in Figure 9. The parity plots and error plots both indicate that this pGP surrogate model with this training was able to well-capture the non-linearity of the data, much better than an OLS multi-linear regression model. Just as the p-GP method can make predictions for compositions not trained on, the p-GP method is also able to make predictions for temperatures not trained on.
0.55 0.6 0.65 0.7 0.75 13.95 14.45 14.95 15.45 15.95 16.45 16.95 Cr Activity Concentration of Cr (Atomic % )
Predicted Actual
30 Table 1: Accuracy of pGP surrogate, accuracy of OLS surrogate, and ratio of their accuracies,*
Fe_a Cr_a Mn_a Si_a C_a Ti_a Mo_a Al_a Nb_a Ni_a OLS Surrogate Average Normalized Error:
5.1 x10-5 2.4 x10-4 8.8 x10-3 8.8 x10-3 2.2 x10-2 8.8 x10-3 8.8 x10-3 4.4 x10-3 8.8 x10-3 4.4 x10-4 pGP Surrogate Average Normalized Error:
2.5 x10-6 1.2 x10-5 4.4 x10-4 4.4 x10-4 1.1 x10-3 4.4 x10-4 4.4 x10-4 2.2 x10-4 4.4 x10-4 2.2 x10-5 Error Ratio:
(Error from pGP)
/ (Error from OLS) 0.05 0.04 0.02 0.02 0.23 0.23 0.10 0.01 0.20 0.01
- In relation to predicting individual elemental activities, denoted by the element symbol followed by _a Figure 7: Accuracy of the modern AI/ML pGP surrogate compared to accuracy of conventional-regression OLS surrogate for elemental activity predictions, as measured by average normalized error relative to the CALPHAD calculation for each component. Elemental activities are noted by the element symbol followed by _a. The left panel shows a bar graph for major components, while the right panel shows a bar graph for minor components. The p-GP surrogate prediction errors (in blue) for the elemental accuracies were typically one order of magnitude smaller than the OLS surrogate prediction errors (in green).
31 Figure 8. Parity plots for the OLS surrogate model predicted activity and actual CALPHAD calculated activity for several elements for 15,000 validation points, for 10 element Febased alloys (Fe, Cr, Mn, Si, C, Ti, Mo, Al, Nb, Ni) at temperatures of 600, 700, and 800 °C. Vertical axes describe model predicted activities while horizontal axes describe CALPHAD calculated activities. In all axes labels, elemental activities are noted by the element symbol followed by _a. There is not good agreement, as seen by comparisons to the 1:1 parity lines.
Figure 9. Parity plots for the pGP surrogate model predicted activity and actual CALPHAD calculated activity for several elements for 15,000 validation points, for 10 element Febased alloys (Fe, Cr, Mn, Si, C, Ti, Mo, Al, Nb, Ni) at temperatures of 600, 700, and 800 °C. Vertical axes describe model predicted activities while horizontal axes describe CALPHAD calculated activities. In all axes labels, elemental activities are noted by the element symbol followed by _a. There is good agreement, as seen by comparisons to the 1:1 parity lines.
32 For safety-related applications, uncertainty estimates of the pGP surrogate model should be trustworthy and within tolerance. Two metrics were examined here for the uncertainties: First, whether
<UGP> was less than UCV (a measure of the GPs ability to include epistemic uncertainty in <UGP>), and secondly whether the final ML surrogate predicted uncertainty (UF) in agreement with the distribution of the prediction errors as measured by the difference between the predictions and actual values (a measure of the final accuracy of UF). For the first metric, looking at all predicted points across all elemental activities, <UGP> was less than UCV for 97% of the values. This suggests that the GPs do a good job at including the epistemic uncertainties in their estimated uncertainties. In the second metric, the 2 x UF was compared to the prediction errors (that is, the residuals, R). For accurate normally distributed errors, 2 x UF < R will be true for 95 % of the points. Here, it was found that 2 x UF < R errors was 95.0 +/- 0.8 % of the time (as averaged across all 10 elements activities and across all 15,000 validation points), which was likely fortuitously good agreement with the expectations. This demonstrates that taking an envelope of 2 x UF as a 95% confidence interval works well. For points found to be further away from the prediction estimate (3 x UF), the residuals were greater than would be expected for normally distributed uncertainties. However, the ML methods used here would allow for more complex uncertainty estimates than what was used here in this limited analysis.
It is appropriate to compare the time of prediction for each composition with the various methods. The CALPHAD calculations take 2-3 minutes per composition with a single processor. For the pGP model, the loading time was on the order of 10 seconds (per cluster), and the evaluation for each predicted elemental activity at a given composition required, on average, 0.0052 s with a standard deviation of 0.0065 s, resulting in < 0.1 s per composition. A multi-linear regression OLS surrogate model was fitted to the same 150,000 points, for comparison. The training and testing for the OLS is orders of magnitude faster, with the model taking up << 1 MB of memory, but it is expected to be less accurate.
Following this training and testing, the activities of 15,000 fresh points (generated within the same range) were used for validation. While the OLS model is the fastest, it is not accurate and not useful. In contrast, the pGP surrogate model is sufficiently accurate and on the order of 1000 times faster than the CALPHAD calculation (Figure 10), while also providing estimated uncertainties that can be utilized to assess when more explicit calculations would be needed.
Figure 10. Comparison of the time per prediction of activity composition using the full thermodynamic calculation (CALPHAD) versus the AI/ML p-GP model. Both calculations were tested using a single central processing unit (CPU) and no graphics processing unit (GPU). The AI/ML p-GP model is more than 1000 times faster than the full thermodynamic CALPHAD calculations. Such technologies will enable AI/ML augmented multi-scale modeling.
33 The demonstration showed that the pGP ML surrogate model has accuracy that is much better than an OLS model. This sparse data during training use case demonstrated that this type of surrogate ML model was suitable for accurate predictions as well as accurate predictions of uncertainties, even when using a reduced, one-dimensional representation of the prediction uncertainties. The uncertainty estimates are thus suitable for sensitivity analysis and uncertainty propagation and can enable reasonable confidence interval predictions for applications where only sparse data is available. These methods are thus suitable for inclusion in multi-scale modeling to overcome data sparsity and will advance capabilities beyond todays routine multi-scale simulations.
4.2 ML Prediction of Metal Alloys Atomic-Scale Interatomic Potential Energy and of Elemental Diffusion Activation Energy In this published example, atomic-scale thermodynamic and atomic-scale kinetic predictions have been made using supervised ML to adjust the penalty term in a regularized linear regression model. As noted earlier, the dissolution and activity of Cr is critically important for corrosion in MSR environments.
Accordingly, predicting quantities that are relevant to the thermodynamics (that is, energetics) and to the kinetics (that is, activation energies/barriers) is of significant importance. A study was recently conducted to demonstrate the ability to use large datasets and regression to make predictions for these quantities in the diverse local configuration environments that Cr atoms can experience inside alloys.134 The study demonstrated the ability to make physics-informed predictions based on large datasets. Thes quantities are applicable to corrosion science of relevance to MSRs. The study investigated the embedded atom interatomic potential energy for Cr as a function of the identity of neighboring elements for thousands of possible local configurations. The bulk-scale compositions checked were Fe70Ni5Cr25, Fe70Ni10Cr20, Fe70Ni15Cr15, and the effects of grain boundaries were checked explicitly for the middle composition. The regression prediction function was trained (or fitted) on >20,000 Cr atoms with different local environments (surrounding atoms) and thermodynamic energies. Some of the Cr atoms used during training were located near grain boundaries.
The thermodynamic predictions from the regression versus the real energy of the training (defined by the embedded atom method interatomic potential) are shown in Figure 11. The model is a physics-informed linear regression model with regularization applied to the penalty term, where the only inputs are the nearest neighbor atom compositions, local electronegativity, and local free volume. There is general agreement in the parity plots, but the error is large. A non-physics-informed model with more inputs could obtain better agreement but would be less physically informative. Here, by using a physics-informed model, the authors were able to assess that the Cr atoms thermodynamic stability is more sensitive to the local electronegativity than to the local free volume, and use of the physically-informed model would likely enable better extrapolation/transferability of the ML model to other geometric environments and possibly even other elements. Additionally, this model may be sufficient to support multi-scale modeling efforts and understand trends, particularly if uncertainty propagation is considered.
34 Figure 11: The regression prediction of the energy of Cr atoms compared to the real energy in (a) 25 (710) bicrystal system with Fe70Ni10Cr20, (b) 5 (310) with bicrystal system Fe70Ni10Cr20, (c) 13 (510) bicrystal system with Fe70Ni10Cr20, and (d) 25 (710) bicrystal system with Fe70Ni5Cr25 as well as Fe70Ni15Cr15. The upper panels represent parity plots. The lower panels represent the density of data. Tens of thousands of total points were used, with greater numbers of points used for the (710) systems. In the first three upper panels, orange points represent Cr located at grain boundaries, and blue points represent Cr located in the bulk.134 In the same study, the authors then attempted to use the same type of ML trained on activation energies to predict activation energies for Cr diffusion to neighboring locations, depending on the local environment. Obtaining training data for activation energies is substantially more computationally expensive, and a smaller training set was used (but still >10,000 points). The ML model was able to have a better ordering of the activation energies for different environments relative to a random ordering. The authors pointed out that this is sufficient to screen the composition space for qualitative comparison of alloys of interest. As an example, the authors note that for picking alloys resistant to high temperature creep, one would desire a system with high thermodynamic stability and low kinetic mobility, and that the ML level of accuracy would be sufficient to assess the desired portion of the phase diagram. Here, by using a physics-informed model, the authors were able to assess that the Cr atoms kinetic mobility is less sensitive to the local electronegativity than to the local free volume, and the use of a physics-informed ML model would likely enable better extrapolation/transferability of the model to other geometric environments and possibly even other elements.
The above example is simply a proof-ofconcept example of large dataset regression applied to a topic of relevance to corrosion with physics-informed ML. It is not an example of state-ofthe-art ML for atomic-scale chemical properties predictions, which would produce substantially smaller errors.
4.3 ML Descriptor Identification and ML Prediction of Corrosion Rates While the processes involved in corrosion are complex, the ability to predict corrosion rates directly based on observed data and physicochemical attributes may be feasible in an approximate way, since the underlying phenomena will follow trends (similar to the concept of the elemental property trends of the periodic table). One promising area of research to mitigate corrosion in environments with halides is to use alloys with four or more elements. The composition space is too vast to be searched explicitly, but directed searches can potentially be accomplished if descriptors can be found that enable prediction of corrosion rates from the alloy composition in conjunction with the conditions of corrosion.
35 Descriptors are a combination of attribute values and equations/models that enable prediction of a quantity of interest using independent variables. A recent ML study attempted to accomplish finding descriptors for alloy corrosion rates in various conditions with aqueous solutions containing halide salts.135 The alloys considered are listed in Table 2, and the conditions had varying pH (4 to 14), temperature (298 K or 313 K), and halide molarity (0 to 1). The study looked at using ML and experimentally measured corrosion rates for 25 compositions of alloys. The intent was that an ML model trained on this data could predict the corrosion rate (in mm/year) for metal alloys in a halide salt aqueous environments at room temperature and could be used to identify multi-dimensional descriptors through analysis of the importance of different features. During initial training, the authors also considered the least absolute shrinkage and selection operator (LASSO), random forest (RF), support vector machine (SVM), but settled on Gradient Boosting Regression (GBR) since it showed a high coefficient of determination, r2, during training and the best r2 during testing. Although the authors did not report confidence intervals for the final predictions, methods with GBR models are capable of providing confidence intervals for predictions. The ML model used was thus GBR, which is a forward feature selection method. The authors began with 32 possible features to include in the descriptor, and then narrowed down the number of features in two stages based on a feature importance criterion that is qualitatively similar to a sensitivity analysis. After these two stages of narrowing, the five features in the final descriptor were (1) the pH of the medium, (2) the halide concentration, (3) the atomic % of the element with the least electrochemical reduction potential, (4) the difference in the lattice constant from an alloy versus a single element phase, and (5) the average electrochemical reduction potential across the individual elements of the alloy. A parity plot for the predictions is shown in Figure 12, along with a comparison bar chart plot. There are several outliers in the plots, which also are reflected in the extreme ratios present in Table 2 (particularly for the alloys FeCoNiCuSn0.04 and Co1.5CrFeNi1.5Ti0.5Mo0.8. The sources of the prediction deviations that lead to outliers could be from a number of factors, such as mistakes in data collection / transcription or lack of information on additional important factors. Although some points deviate from the parity line, it is clear that the model is able to estimate corrosion rates with order of magnitude accuracy, and that the identification of important factors for prediction was successful. As can be seen, there is an over-estimation of rates among the lower corrosion rates, such that the estimates are conservative in this example. This example demonstrates that machine-learned feature selection can be used to gain insights into the most important features, to create descriptors, and to ultimately create a model.
36 Table 2: Experimental and Predicted Corrosion Rates in mm/year for 25 Alloys under varied conditions, from ref 135.
The ratio of the predicted rate over the experimental rate is shown in the last column.
Alloy Experimental Rate (mm/year)
Predicted Rate (mm/year)
Predicted /
Experimental Al2CoCrFeNi 0.02965 0.06498 2.191568 Al2CrFeNiCoCuTi1.5 0.03558 0.04256 1.196178 Fe68.59Ni10.47Co0.21Mo2Cr16.61 0.00115 0.00299 2.6 Al2CrFeCoCuTiNi 0.10533 0.09908 0.940663 Fe24.85Ni25.89Co26Mn0.51Cr22.66Al0.07 0.00803 0.01394 1.73599 Fe46.86Ni12.88Co12.54Mn11.72Cr15.8Nb0.16 0.00959 0.00339 0.353493 FeCoNiCu 0.01561 0.12714 8.144779 FeCoNiCuSn0.02 0.01177 0.01977 1.679694 FeCoNiCuSn0.03 0.01043 0.01977 1.895494 FeCoNiCuSn0.04 0.01532 0.01977 1.29047 Co1.5CrFeNi1.5Ti0.5Mo0.1 0.00121 0.00307 2.53719 Co1.5CrFeNi1.5Ti0.5Mo0.5 0.00178 0.00414 2.325843 Co1.5CrFeNi1.5Ti0.5Mo0.8 0.00355 0.00707 1.991549 Co1.5CrFeNi1.5Ti0.5Mo0.1 0.00103 0.00229 2.223301 Co1.5CrFeNi1.5Ti0.5Mo0.5 0.00143 0.00229 1.601399 Co1.5CrFeNi1.5Ti0.5Mo0.8 0.00216 0.38237 177.0231 Al2CrFeCoCuTiNi0.5 0.25814 0.35711 1.383397 Al2CrFeCoCuTiNi1 0.10533 0.09908 0.940663 Al2CrFeCoCuTiNi1.5 0.52126 0.33121 0.635403 Al2CrFeCoCuTiNi2 0.54881 0.33803 0.615933 FeCoNiCuSn0.02 0.02380 0.02088 0.877311 FeCoNiCuSn0.03 0.03611 0.02088 0.578233 FeCoNiCuSn0.04 1.08367 0.02088 0.019268 FeCoNiCuSn0.05 0.01510 0.03004 1.989404 FeCoNiCuSn0.07 0.02082 0.03004 1.442843
37 Figure 12. (Upper) Log-scale parity plot for the GBR model prediction values versus experimentally measured values for corrosion rates which were measured and predicted for mm/year. (Lower) The predicted rates compared to the measured rates, shown as log-scale bars and sorted by increasing corrosion rate.
4.4 Integration of Atomic-Scale ML Predictions into Multi-scale Modeling Methods Once the atomic-scale salt-metal interactions are described, the mesoscale thermodynamics as well as diffusion of dissolved alloying elements, corrosive species, and salt constituents in the alloy microstructure can be elucidated based on these atomic-level simulations. Both species diffusion in the confined liquid salt and species diffusion along the alloy surface must be considered. Recent phase-field modeling shows136, some of these diffusion coefficients are critical input parameters for modeling the morphology evolution during a dealloying process.
Complex chemical reactions and species diffusion can occur at the alloy/salt interface as well as at alloy grain boundaries (with possible chemical species encompassing corrosion products, oxygen containing impurities, salt constituents, and the alloy elements).. These processes can be rate-determining in molten salt corrosion. Further, species mass transport plays a critical role on governing the morphology evolution of the alloy/salt interface during the corrosion. Recently developed neural network 0.001 0.01 0.1 1
0.001 0.01 0.1 1
Prediction Experiment 0.001 0.01 0.1 1
10 Corrosion Rate (mm / year )
Alloys Sorted by Increasing Corrosion Rate Measured Rate Predicted Rate
38 interatomic potentials (NNIP)137 can be employed to study the absorption of corrosion species, dissolution of alloying elements and oxides, species diffusion at the alloy/salt interface of different surface crystallographic orientations and alloy compositions. These high-throughput NNIP-based molecular dynamics (MD) simulations will enable a rapid evaluation of property data such as thermodynamic and thermophysical data to construct equilibrium 2D/3D phase diagrams in the CALPHAD framework and surface adsorption isotherms. The results will not only provide atomic-level insight for interpreting our experimental observations, but also be used as input for the mesoscale modeling of the microstructural evolution at the alloy/salt interface.
For the species diffusion in the confined liquid salt, MD simulations with large numbers of molecules (hundreds to thousands) using accurate interatomic potentials are essential, and the only path known to achieve such simulations at this time are the NNIP-based MD simulations. These atomic-scale modeling methods will also allow the consideration of the effects of surface orientations and alloy compositions. The lower spatial scale models can be interfaced with rapid high-fidelity reduced order models to predict the compositional changes and phase transformations in the alloys as a function of time, temperature, salt chemistry and alloy composition. A schematic of such a combined approach is shown in Figure 13.
Figure 13. Schematic of the AIdriven multi-scale modeling research Even if the elementary step chemical kinetics are known completely, there will still be stochasticity at the continuum (macroscopic) level. The macroscopic stochasticity will arise due to atomic-level stochastic processes138 and inherent concentration heterogeneities (including due to impurities, defects and microstructure). These sources of stochasticity result in non-uniform dissolution rates for bulk materials,74 65, 139 as can be seen in Figure 4. The concentration profile as a function of depth shows kinks in Figure 4, and these kinks would not be present if a simple uniform rate dissolution model were applied to a material with an initially homogeneous distribution of concentrations. For example, grain boundaries can give rise to such kinks in the macroscopic chemical kinetics.65, 74 Ultimately, regardless of its atomic-scale origins, localized corrosion in aqueous environments, due to pitting and crevices, results in a non-uniform depletion of material during corrosion.140 The local corrosion rates at such locations can be many
39 orders of magnitude greater than the non-localized corrosion rate.141 Localized corrosion is presently not fully understood, likely depends on local concentrations and arrangements of elements,142, 143 and is complicated by the rate also being dependent on the nature and concentrations anions in the aqueous phase.140, 144 While the atomic-scale mechanisms of corrosion are not completely understood, there have been measurements of pitting resistance among many alloys, giving rise to empirical linear descriptors that can predict a scalar value for pitting resistance after fitting within a group of compositionally similar alloys, and the predicted scalar value is called the pitting resistance equivalent number (PREN).144, 145 Recognizing the utility of these empirically derived relationships, a database of electrochemical metrics for corrosion resistant alloys was created, which is a starting point of data for DS/ML analysis.140 ML approaches are already being utilized to predict phenomenological corrosion rates based on phenomenological data.141, 146 It is likely that even this phenomenological level of data, if the database is made large enough and include microstructural information, will be able to provide important physics-informed ML descriptors and finally physically meaningful equations.147 To do so, it is critical to obtain large enough databases (such as thousands of experiments), and high-throughput electrochemical studies have been proposed for this purpose.39 In practice, considering the stochastic nature of local corrosion, there will be a range of rates observed when an experiment is repeated with new samples. For full MSR modeling, more physically realistic parameter values can be obtained by considering the uncertainties from experiments, chemistry-based simulations,74 and nuclear physics-based simulations148 using the DS method of Bayesian Parameter Estimation (BPE).149, 150 BPE enables extraction of the most physically realistic estimates of parameters (such as activation energies). For multi-scale physics models such as in corrosion, parameter estimation and uncertainty quantification can require large numbers of simulation evaluations (sometimes much greater than 103 evaluations) and surrogate models may be required to enable computational tractability to be achieved.
4.5 ML Augmented Continuum-Scale Modeling (Fluid Dynamics and Heat Transfer)
As noted earlier, another use of ML augmentation in modeling is to use ML surrogates (or ML error correction) for computational fluid dynamics simulations (see provided references).36, 151-155 Creating surrogates for time dependent partial differential equation based models can be challenging. Dynamic mode decomposition156 is a relatively new data-driven methodology that has shown success in creating reduced order surrogate models for such problems and has already been applied to modeling of a molten salt reactor core.157 Reduced order models are also important because they would be needed for real-time digital twin multi-scale physics modeling of nuclear powerplant operation,158 and this would enable superior real-time safety technologies when combined with real-time monitoring. The literature includes significant advances in tools to create continuum-scale ML surrogate models, ML predictions, and ML error correction. These advances will enable more precise long-term simulations as well as potentially real-time digital twin simulations, as the technologies become more mature.
5 Key Observations To fully utilize the new predictive AI/ML/DS capabilities becoming available to achieve more precise long-term net corrosion rate predictions will require a dedicated effort to collect appropriate training data. This effort would also require coordination between different types of researchers and developers. Accordingly, we provide some possible courses of action that would be needed for the field to fully realize the potential of these new methods.
40 5.1 Long-Term Intrinsic Corrosion Rates Differ from Short-Term Intrinsic Corrosion Rates.
Intrinsic corrosion rates would be constant at a given temperature if the system had perfect mixing and the surface area, surface alloy composition, and salt composition did not change over time.
However, in practice there are dynamic changes in surface area, mass transport, and concentrations in both the fluid and at the surface of the solid. The alloy and molten salt combinations currently being considered seriously for use in MSRs tend to have a fast initial corrosion rate followed by a slower long-term corrosion rate. The early time corrosion rate is faster than long-term corrosion rates due to (a) impurities in salts (not focused on in this report), and (b) changes in the concentration of elements in the alloy at the surface and near-surface from leaching corrosion (more focused on in this report). Grain boundaries play a role in the mass transport of elements out of the alloy for leaching corrosion, especially in the short-term corrosion rates. The slower long-term corrosion rate is believed to be largely rate-controlled by slower long-term diffusion of elements out from the alloy (which is different from passivation,). For example, in the work by Pillai et al.62 with KCL-MgCl2 salt and alloys containing Fe, Ni, and Cr it was observed that the corrosion rate was relatively rapid at up to ~500 hours but that the rate had stabilized to a lower corrosion rate before ~1600 hours. At present, it is not possible to predict the long-term corrosion rates from short-term corrosion rates, and it is not feasible to collect long-term corrosion rates for all materials and all salts. AI/ML combined with physical modeling may be able to bridge this prediction gap. The data used for training AI/ML models could potentially involve data from accelerated corrosion testing.
5.2 Acquisition of Relevant Experimental Data for Databases A common conclusion that can be drawn from the previous sections is that there is a need to gather additional relevant data over longer timescales to enable development of reliable predictive models. Long-term experimental measurement acquisition is costly, and it is clear that such datasets will be sparse and may need to be carefully chosen as there is a limited understanding of rate governing material degradation processes. If the approach proposed by Pillai et al.71 is considered, the measurement of dissolution rates of the pure elements found in Fe-and Ni-based structural materials (such as the elements Cr, Al, Ti, Mn, Fe and Ni) in technically relevant salt mixtures as a function of temperature was sufficient to describe most of the corrosion-induced microstructural evolution in multicomponent and multiphase alloys - at least for the case of using purified molten salts in capsules (static testing). The experiments to measure dissolution rates of pure elements involve exposure of specimens of the elements in the corresponding molten chloride and fluoride salts in relatively inert capsules for durations equivalent to the times required for saturation of the salt. The typical exposure times for these experiments are of the order of 3-5 kh and specimens are removed at predefined time intervals to allow the estimation of the dissolution rates. Post-exposure, the salt chemistry is analyzed to measure the content of the dissolved element and rate constants are estimated. These rate constants can then be implemented as in models to predict the compositional evolution in the alloy. Using this approach, an excellent comparison between measured and calculated Cr depletion in both Fe-and Ni-based alloys exposed in the purified binary KCl-MgCl2 eutectic salt (static testing in quartz capsules) was achieved. The applicability of this approach to flowing tests remains to be ascertained. Validation of the models aiming to predict corrosion-induced material lifetimes will require performing experiments in conditions replicating actual reactor operation as best as possible (e.g., salt chemistry, flow conditions, temperatures, timescales). These experiments will additionally need to quantitatively address the role of impurities (i.e.,
O2, H2O, OH-, and H+, metallic inclusions such as Ni, Fe) in accelerating corrosion or the influence of additions for redox control (e.g., Mg, Be) to suppress corrosion. To conclude, a simple augmentation of
41 experimental datasets to account for the impact of all key parameters governing corrosion in molten salts is expensive and might not be essential. The resources should focus on generating a higher number of datapoints that will represent sufficient variations in variables of interest, based on chemical intuition and existing physics-based models. To create these datasets, new investments may be needed in high-throughput experiments, potentially enabled by advances in robotics and automation (which itself is advancing in part due to ML methods). Accelerated corrosion testing may play a role. Once such data is collected, ML methods such as LASSO or SISSO can be used to identify the most important features, to gain insights, and to help create either accurate or highly explainable surrogates, depending on the application.
5.3 Need for Combined Efforts Between Modeling and Experimental Studies.
The previous subsection noted that LASSO, SISSO, and related methods can be used to identify the most important features of a system to predict corrosion rates. To gain the most information, and to best utilize this information requires combined efforts between modeling and experimental studies.
Modeling can identify which experiments will be most informative, and accuracy of the refined models must also be benchmarking against experiments. There are two reasons that benchmarking is needed. 1)
The governing physical equations are not always known, and a feedback loop with combined efforts of modeling and experiment can be used to shed light on governing phenomena (such as grain boundaries).
- 2) Even when the base physical equations are known, there will always be non-idealities. This means that empirical error correction terms (whether using modern ML methods or other approaches) will be needed to achieve the most accurate predictions. Within such a feedback loop, Bayesian DS methods will enable improved knowledge of physical constants and values as more data is collected.
5.4 Increasing Focus on Sources of Uncertainty/Variation/etc.
Uncertainty quantification (UQ) for simulations needs to be an essential part of model development for molten salts corrosion processes (characterization of uncertainties and their propagation). Some of the expected sources of uncertainty classified based on the type of data are tabulated in Table 3.
The experimental variability in the measured dissolution rates needs to be accounted for since this uncertainty will propagate through the computational models. The influence of uncertainties in the dissolution rates measured experimentally as well as those implemented in the coupled thermodynamic-Table 3. Expected sources of uncertainty Experimental Computational Experimental variability in measured mass change data (input for models)
Experimental variability in measured dissolution rates (input for models)
Variability in salt chemistry (e.g., impurity content)
Differences in capsule materials (static tests)
Variable Volume/Surface Area ratios Sources of inhomogeneity in the materials (impurities in the alloys, grain boundaries)
Uncertainties in thermodynamic and kinetic data Sensitivity of the model to initial conditions Modeling error from evaluation of ordinary (or partial) differential equations Modeling errors from approximations to physical equations to make problems computationally tractable
42 kinetic model will need to be quantified. Sensitivity analysis for the physics-based thermokinetic models with respect to the uncertainties in the input dissolution rates will need be analyzed to provide information on how the subsequent incorporation of these quantities, The use of AI/ML driven approaches, which will greatly accelerate the simulation workflow, would require a rigorous characterization and propagation of the uncertainties in this dataset. It is important to use ML models that either inherently include uncertainties or are capable of having complementary uncertainty models trained alongside the primary model. Within the workflow, information will need to be combined from disparate experimentally measured datasets with varying sources of uncertainty. These datasets will need to be analyzed separately and also as a whole for hierarchical uncertainty analysis to include the joint uncertainties, likely with Bayesian methods. These uncertainties can then be propagated in the ML training process. Following this UQ characterization, the uncertainty of the ML predictions will be included in the modeling for a correct propagation into the multi-scale modeling of the chemistry and physics.
5.5 ML Augmentation Should Be Gradually Incorporated into Predictions.
As indicated earlier, there are several portions of AI/ML/DS that are mature or are being actively developed for applications that are relevant for MSRs. The technologies which are maturing and which should be gradually incorporated into modeling are 1) accurate but non-physical sparse data surrogates with inclusions of uncertainties, as this can be used for increasing the efficiency in multi-scale modeling,
- 2) machine-learned ordinary and/or partial differential equation evaluation surrogates and error correction, as these can increase both efficiency and accuracy in computational fluid dynamics and related types of simulations, 3) symbolic regression, dimensionality compression, and explainable AI should be utilized in research applications to more easily understand what factors are important in predicting physical observations that are not completely understood, including the effects of minor factors.
- 6. Summary There has been a resurgence of interest in MSRs. Corrosion resistance of structural materials is important in the technical realization of MSRs. The current state of knowledge provides sufficient confidence that corrosion resistant technologies exist, such that appropriate structural material and salt choices should enable safe use of MSRs. Although such corrosion resistant technologies exist, there would be a benefit from having improved models to predict the long-term corrosion rates. There are knowledge gaps in understanding of the governing mechanisms, and correspond gaps in our ability to quantitively predict corrosion rates in corrosion-resistant alloys. Enhanced quantitative understanding of extent and degree of materials compatibility with harsh salt and radiation environments, to predict long-term (years-scale) materials degradation and failure, would be helpful. Todays experiments, modeling, and monitoring for short timescales (minutes, days, and even months) are not fully capable of enabling the quantitative predictions that future modeling is likely to achieve. The early time corrosion rate is faster than long-term corrosion rates due to (a) impurities in salts (not focused on in this report), and (b) due to depletion of elemental concentrations in the metal solid surface and near-surface, due to leaching corrosion (which is more focused on in this report). After the early corrosion rates, the slower long-term corrosion rates are believed to be largely rate-controlled by diffusion of elements out from the alloy. At present, it is not possible to precisely quantitatively predict the long-term corrosion rates from short-term corrosion rates, and it is not feasible to collect long-term corrosion rates for all materials and all salts.
43 AI/ML combined with physical modeling may be able to bridge this prediction gap. It is vital to develop, validate, and verify, computational simulation methods and models that can be used to predict materials compatibility and performance for MSRs over operational timescales. New methods and additional data can help with this goal. The data used for training AI/ML models could potentially involve data from accelerated corrosion testing.
There are several areas that we have identified where AI/ML/DS can help to close prediction and knowledge gaps for precise, quantitative long-term corrosion predictions for MSRs. These include, but are not limited to a) making predictions about the properties and behaviors of chemical compositions and performance under various conditions, through either physics or non-physics-informed ML, b) the ability to extract which factors and equations functional forms can describe a phenomenon well, with capabilities beyond that of simple sensitivity analysis, c) the ability to cluster and classify data to predict and understand trends, d) the ability to fill data gaps by AI/ML predictions from sparse data with higher accuracy and better uncertainty estimates than simple interpolation, e) the ability to accelerate differential equation and finite element based modeling, to enable more computationally tractable fluid dynamics, heat transport, and chemical kinetics modeling. Section 4.1 showed a demonstration of how data gaps can be filled by AI/ML/DS for multi-scale simulations, and Section 4.3 showed an example of how predictions can be made of corrosion rates directly from sparse experimental data and AI/ML/DS.
As shown in this report, these technologies do have promise for predicting chemical activities for various alloy compositions, predicting corrosion rates, and providing physical insights. The current progress is encouraging. To fully utilize the capabilities that are becoming available, we have provided a set of Key Observations in Section 5, including which types of data and collaborations are needed to fully utilize possible technological enhancements and synergies. Realizing the full potential of AI/ML for these applications would require additional experimental data collection (to be obtained with dedicated coordination between experimentalists, modelers, and AI/ML practitioners) as well as further development of the AI/ML methods.
44 Appendixes Appendix 1: Explanation of Various AI/ML/DS Methods With the earlier provided definitions in mind, we can briefly review some concepts and the state-ofthe-art in AI, ML, and DS, as well as present challenges and opportunities as related to applications of relevance to MSRs. It is important to recognize that there are a range of tools available, which will each have different strengths and weaknesses. ML is divided into unsupervised learning, supervised learning, and semi-supervised learning.36 Unsupervised learning refers to cases where the ML does not require any human input other than the data and the objective: it can be used for clustering data, finding associations between data, and dimensionality reduction of data. In this context, dimensionality reduction of data involves creating approximate forms of the data with the intent to minimize inaccuracies while balancing a reduction in the size and complexity of the data. Supervised learning requires the machine to receive already labeled datasets (that is, to know the answers or to be able to calculate the answers for an initially provided dataset). Generally, supervised learning is used for classification/decision problems as well as prediction from regression. Semi-supervised learning involves the machine receiving external feedback about whether it has completed the task correctly or not and then improving its model accordingly: this includes reinforcement learning from either a human or another program that is evaluating the outputs.
Supervised and semi-supervised learning are typically used for prediction problems, while unsupervised learning is typically used for gaining understanding from data. Supervised and semi-supervised learning can create high accuracy models of datasets but have an upfront (or continuing) cost of labeling or otherwise generating answers for training. Unsupervised learning typically requires larger datasets than supervised ML and is less suitable for making accurate models of datasets. From these distinctions, we can recognize that ML relies upon DS. Yet, DS is also complementary to ML: some DS insights can be gained without ML, and those insights can be used to guide or inform ML practitioners. In the context of AI that is based on ML, we can say that AI describes the ability of the program to provide useful outputs after ML is finished. Thus, AI/ML/DS are often not totally separable.
To provide context for the classes of ML learning described above and how they might be useful in MSRs, it is good for physical simulation practitioners to know what typical applications are.
Unsupervised learning is primarily used for (a) naive groupings of data (kmeans clustering, spectral clustering) and/or (b) dimensionality reduction / compression of the data (principal components analysis, proper orthogonal decomposition, autoencoders, and diffusion maps). These methods typically require the most amounts of data points, and can be used to help gain insights, efficiency, or to pre-process data prior to supervised learning. Supervised learning is used for classification and quantitative prediction problems. The division of the methods and applications (classification versus quantitative prediction) is less clear for supervised learning than a learning reader might expect because application of binning methods and discretization methods to the solutions space enables classification methodologies and quantitative prediction methodologies to be used for the other purpose, respectively. Supervised learning techniques include linear regressions, neural networks (including deep learning), gaussian processes, decision trees (including random forests), support vector machines, static model Markov decision processes, knearest neighbors classification. Semi-supervised learning (including reinforcement learning) is typically used for classification and for problems which rely more on logic and previously unseen stimuli.
Semi-supervised learning is distinguished from supervised learning in that only a small number of initially labeled data is required to initiate the training, and a full retraining would not be required when more data is added. Semi-supervised learning techniques include some neural networks (such as deep reinforcement learning), generative adversarial networks, and reinforcement model Markov decision processes. It is important to recognize that there is no distinct separation between machine learning and multi-parameter regression: modern ML has shown gains in technology by use of many-parameter
45 models, but the definition of ML we use here also includes simple linear regression. There is no dividing line in the space between a two-parameter linear model and a million-parameter complex-function model - only increasing levels of model size and complexity. Similarly, one can say there is now no line between physics based models and phenomenological models: there is a continuum through the use of hybrid models due to ML/AI augmented physics-based models (which can be more accurate than models that are not augmented). Each of the types of ML can be useful for MSR modeling, with applications ranging from parameter value extraction, direct prediction, or for simulation. Some examples will be provided in this report. Many of the methods presented were known 50 years ago but have become of greater usefulness today due to the present-day computer processing power, computer memory, and accumulated databases.
In ML augmentation for simulations (for both physics-informed and non-physics-informed ML),
there are four principal categories: (a) surrogate prediction of static/scalar quantities, (b) surrogate prediction of dynamic / time-series quantities, (c) error correction for scalar or time-series predictions, and (d) ML based differential equation solvers. The first two of these categories are now mature technologies, with broad ranges of accuracy achieved depending on the extent of data available and the shape of the response surface. Importantly, ML can be used to fill gaps in data. Surrogate models (including interpolation) can be particularly useful when the underlying physical phenomena are not completely understood, or too computationally expensive to simulate routinely, or when only sparse accurate data is available. Surrogate prediction of dynamic / time-series quantities is typically used for achieving reduced computational cost. For example, chemical kinetics simulations have been successfully approximated by random forest surrogates,159 and advanced interpolation methods have enabled cheaper computation of complex terms in multi-scale modeling.21, 160 Error correction of scalar or time-series simulations (including fluid dynamics) has proven to be a very effective strategy in recent research.
Cheaper simulations through ML present a significant opportunity for MSR applications: as noted earlier, fluid dynamics simulations can become unfeasibly expensive, especially for long-time materials behavior.161, 162 Methods enabling cheaper simulations with error corrections have been applied to fluid dynamics simulations.36, 163 ML error correction can additionally enable improved accuracy when built upon existing conventional state-ofthe-art simulations or through the use of simplified models. The combined advantages of surrogate models and error correction have enabled significant strides in the accuracy and accessibility of atomic-scale and multi-scale simulations. These capabilities should make long-term simulations feasible that are presently not possible. Complementary to simulations, AI/ML/DS have a proven ability to detect anomalies, which may be used to identify early warnings of failure.164 Even for cases that are less extreme than failure warning, online monitoring coupled with ML can be utilized for adaptive responses to improve load management.165 Both failure warning and adaptive responses can be improved by including not only measurements but simultaneous (real-time or asynchronous) physical simulations37 for comparison to the measured signals.
One of the state-ofthe-art technologies for achieving physics-informed ML is by use of the SISSO.166, 167 SISSO is a compressed sensing operator designed for use with a huge features space (that is, rather than only using two features, one might provide hundreds as inputs). SISSO then optimizes the coefficients for the features and mathematical operators to find an optimal descriptor formula. As an example, SISSO was able to assess that 2-element materials could be classified as metals or non-metals when using an X axis of
/
()
where and are the atomic compositions of the two elements, and are the Pauling electronegativity, IEB is the ionization energy of the second element, and / is the packing fraction.167 Using this physics-informed ML model, the authors were then able to correctly predict when
46 a non-metal would convert into a metal as a function of pressure,168 which impacts. SISSO has been shown to be powerful at finding relationships between features and properties that are physically meaningful and accurate for a number of systems.167-171 When applied to sufficiently large datasets which are appropriately annotated, SISSO and any similar technologies will be transformative in the ability for research to find accurate and physically meaningful relations. For the example MSR application shown above, it is not clear that the regression model used would perform better than uninformed sparse data interpolation, but with enough features, the physics-informed ML model can facilitate insights and extrapolate beyond the type of local atomic environments on which it was trained. Neural Networks, Random Forests, and other non-physical forms of machine learning can outperform direct physical modeling with gains in accuracy and a loss of physical meaning, but it is possible to use such ML methods with SISSO guidance during training to develop physically meaningful ML models.172 The use of compressed sensing methods such as SISSO, in conjunction with ML, will enable fast, efficient, and accurate predictions of atomic/molecular-scale properties of interest. Additionally, the use of compressed sensing and related methods for symbolic regression are capable of finding physical governing equations.173 These methods will thus also provide the ability to extract physical equations that might describe causes of deviations from ideal simple equations.
An important part of long-term physical model accuracy improvement is to obtain ever increasingly accurate physical models (simulation functions) as well as physical parameters for models (such as activation energies). BPE is a powerful method from DS that has the capability of utilizing ML predictions and experimental data to extract the most physically accurate estimates of parameters from measured data.149, 160, 174, 175 As noted in the references, two of the advantages of BPE are that it can provide credible intervals (uncertainty estimates) for the final parameter estimates, and that both the parameter estimates and their uncertainty estimates can be updated easily when new data is available (it does not require construction of a new model, which is another advantage relative to regularization methods).
A complicating factor for simulating MSR behavior is that, in many cases, real world behavior of some phenomena (such as materials failure) is not dictated by solely deterministic processes (at least, it cannot be modeled as deterministic at the macroscopic level given the information available). Instead, the real-world behavior may be dictated by apparently stochastic processes where there is a certain probability of an event to occur. The only way to model the long-term behavior of such cases is to include uncertainty envelopes for when certain events might happen. Of course, the model can involve a feedback loop to update the uncertainty of the long-term behavior based on the newest monitoring information available. On the simulation side, such stochastic processes are simulated using Monte Carlo methods or trajectory sampling methods.162, 176, 177 Obtaining the uncertainty envelope (or even the expected mean behavior) can require large samplings of possible trajectories, which is sometimes extremely expensive to evaluate. Accordingly, various methods have been developed to accelerate Monte Carlo simulations162, 176-180 as well as for trajectory sampling.181-183 For stochastic behavior system predictions, the uncertainties in the predicted outcomes are important, and this must be considered when surrogates are used for such systems. ML surrogates can include uncertainty propagation, with separate terms for the uncertainties of the predicted outcomes. There is currently a rise in development in uncertainty propagation of ML based modeling, and this rise will increase the usefulness of ML surrogates for applications where stochastic behavior uncertainty envelopes are needed (including corrosion and radiation-based materials failure).
Appendix 2: Explainable / Interpretable AI While applying AI/ML to nuclear applications, it is critical that reliability and safety implications be considered.184-193 The blackbox nature that often accompanies ML is considered a challenge for the
47 nuclear industry, because it inhibits practitioners abilities to ensure that AI/ML models are built in a way that avoids giving unrealistic outputs. The desire for reliability of results and avoidance of high-error outputs is integral for safety, but also sought after on non-nuclear applications. For many applications, if a practitioner attempts to assess how (or why) the model might choose outputs/behaviors under real world conditions, the practitioner desires to have trust in what the assessment produces. Accordingly, there is work in AI/ML research to create interpretable or explainable AI.14, 194, 195 The terms explainable and interpretable AI are often used interchangeably.196 The U.S. Defense Advanced Research Projects Agency (DARPA ) chooses to use the term explainable AI rather than interpretable AI to reflect DARPAs desire for human-understandable AI.194 DARPA defines194 explainable AI as AI systems that can explain their rationale to a human user, characterize their strengths and weaknesses, and convey an understanding of how they will behave in the future. As noted earlier, many AI/ML models are based on forms of regression, especially for cases of relevance to MSRs since the quantities of interest are usually numerical. Regression based ML involves optimizing internal optimized parameters (including hyper parameters) which may be many more in number than the known physical parameters, and there may be no obvious relation between the models internal parameters and known physical parameters. It is important to recognize that in this context, the optimized model may not be the best fit to the training data, since a models purpose is generally for predictions that go beyond the training data, or for gaining insights (and the computational efficiency for training or for evaluating the model may also be important).
Predictive accuracy is related to how well the model can predict new data, while descriptive accuracy is related to being able to interpret (or explain) what the model has learned. There are two broad methods by which this can be achieved: there is model-based interpretability in which the model is constructed with the intent to be interpretable, and there is post-hoc interpretability in which a trained model is analyzed to gain insights into relationships that exist in the trained model.195 When models are physics-informed ML, then there is clearly model-based interpretability. Model-based interpretability often involves using simpler but less accurate models, while post-hoc analysis can be applied to arbitrarily complex models.195 Model complexity can impede the ability to interpret any observed relations, and an important aspect of post-hoc analysis can be visualization and/or other human-interpretable sensitivity analysis.195 One issue which must be considered for both physics-informed ML and other models is to recognize that a particular parameter or feature may be the most important across the dataset (that is, a type of global sensitivity) but that for various points or regions of the dataset other parameters may be most important (that is, a type of local sensitivity).195 Recognizing that local sensitivity and global sensitivity may provide different assessments of which parameters are most important, we can consider some strategies that have been used for creating explainable/interpretable models.
An overview of strategies has been provided in reference196, and all of the strategies noted below are described within this reference. The strategies can be grouped into several themes. We first start with the themes for post-hoc analyses. 1) A simple functional form can be used intentionally, or a local surrogate can be constructed that is simpler than the full model in order to gain insights (this is similar to using piecewise simpler functional forms), 2) Individual datapoints or sections of the parameter space can be classified based on ifthen rules, such as if an atom has an atomic weight less than 20, then it is not a transition metal, 3) If the explainer/post-hoc analysis software has access to the internals of the model, then the explainer can decompose the different layers of the model (such as by Layer-Wise Relevance Propagation or Deep Taylor Decomposition). It is worth noting that Layer-Wise Propagation can be used to help remove unrealistic predictions by identifying the most important (and thus most relevant) portions of an underlying model. This can, for many systems, remove minor terms that would otherwise have extreme behavior in untrained conditions. 4) Direct sensitivity analyses, such as Prediction Difference Analysis, Testing with Concept Activation Vectors, Integrated Gradients, and Meaningful Perturbations.
There are also several themes among model-based interpretability strategies. 1) Explainable Graph Neural Networks involves rewarding and penalizing desired behavior based on domain-specific knowledge during
48 the initial training. 2) There are strategies which either start with a large number of variables and reduce them to the most important, such as the SISSO, as well as strategies which start with no model and involve only adding parameters that are the most important, such as various Shapley value-based methods. 3)
Textual explanations involve using a second AI that is trained on/with the main AI/ML to provide a textual explanation that can be read by a human for any prediction - this involves a domain expert assessing outputted explanations to see if they are reasonable. 4) Explainable Neural Symbolic Learning involves using prior human knowledge to construct the model (it is an explainability by design approach, while still using functions that are potentially more complex - or at least different from - the true governing equations).
One must be careful to recognize that post-hoc interpretations are always based on correlations and not causation. Further, even model-based interpretability can be misleading. In all cases, one can say that a certain input caused the ML to create a certain output, but this is not the same as physical causation.
Trained data may be tested, but it is not a substitute for hypothesis-based testing. Rather, it gives insights into relations of physical quantities, from which a hypothesis may be made. For many applications, the relations are sufficient even without knowing if the relationship is due to causation rather than simply correlation. For other applications, if fundamental physical understanding is needed, hypothesis testing will still be required. Expert judgment can be needed to assess whether hypothesis testing should be conducted. There are some cases (such as ML approximations for ordinary differential equations) where the technology can be used widely generally once subject matter experts (such as ordinary differential equations experts) have confirmed that the training is appropriate to use beyond its training.
Figure 14. Learning Performance Versus Explainability Tradeoff for Several Categories of Learning Techniques.
Reproduced from reference 194.
There is a tension between explainability and accuracy, as shown in a qualitative way in Figure 14.194 Through both model-based interpretability and post-hoc analyses, the points in Figure 14 can be moved further to the right in the graph points, while still following the same original trend. In 2021, the DARPA Explainable AI Program came to its end, and a retrospective was published.197 It was noted that there is no universal solution to explainable AI as different users need different levels of explanation.
However, it was found that explainability did indeed help to avoid high-error edge cases, and that advisability (the ability for the user to teach the AI when it is incorrect) increased user trust. The current generation of explainable AI is still in its early stages, and one can be optimistic that there will be
49 technologies developed which find a better balance between performance and explainability and show enough utility to become widely adopted. Many software packages are already being developed for explainable AI (see for example the lists in 196, 197).
Appendix 3: Details of Piecewise Gaussian Process Based ML Surrogate Model To create training and testing data, CALPHAD was used to calculate the thermodynamic activities (of each of the elements present) for 50,000 points of Febased alloys (Fe, Cr, Mn, Si, C, Ti, Mo, Al, Nb, Ni) at temperatures of 873.15, 973.15 and 1073.15 K (600, 700, and 800 °C), with the points chosen by astroidal sobol sampling in the realistic concentration ranges. The ranges that were used are shown in Table 4.
Table 4. Atomic Composition Range for Training and Validation (in Atomic Percentages)
Fe Cr Mn Si C
Ti Mo Al Nb Ni Min 37.5 0
0 0
0 0
0 0
0 0
Max 100 25 1
1 0.5 2
1 4
3 25 Central 86.3 18 0.5 0.5 0.2 0.5 0.5 1
0.5 10 The CALPHAD calculations additionally provided the percent of FCC-gamma phase and BCC-Alpha phase, which were also used in the training. The activity of the elements is not simply a linear function of their concentration, and the presence of other elements affects the thermodynamic activity of each other.
Figure 15 shows each elements activities in this range (sliced out of the multi-dimensional space) to illustrate that the activities do not simply have a functional dependence on the concentration of those elements. Clearly, the response is more complex than linear, and predictions will not simply be a function of that elements concentration. This indicates that there is a multi-dimensional problem which may be non-trivial to create a surrogate for.
50 Figure 15. The activity versus the concentration for each element for training points, as calculated by CALPHAD for Febased alloys (Fe, Cr, Mn, Si, C, Ti, Mo, Al, Nb, Ni) at temperatures of 600, 700, and 800 °C (1112, 1292, 1472
°F). The activities do not simply have a functional dependence on the concentration of the respective elements.
Although there are 150,000 points total, this is a sparse dataset problem because even 150,000 points is insufficient to accurately interpolate with simple linear interpolation for this 13-dimension, nonlinear dependence. However, as will be shown, a ML surrogate model will be able to predict the activities of elements with high accuracy (relative to the CALPHAD model) and good computational efficiency. There is an upfront training cost, but following the training, the ML surrogate model enables predictions at arbitrary compositions and temperatures within the range of training.
The ML surrogate model methodology that was chosen is based on Gaussian process (GP) regression. GPs have gained popularity in ML due to their combination of high accuracy and an ability to provide estimated uncertainties for predicted points. GPs are less prone to overfitting sparse data relative to NN.22, 198 However, GPs generally do not have explainability for their predictions: their strengths are best utilized when accuracy and uncertainty predictions are prioritized over explainability. A limitation of GPs is that they do not scale well to large datasets and training points with cubic scaling cubic scaling O(n3) in the number of computations required relative to the amount of training data.22, 199 Due to GP scaling, divide and conquer techniques such as bagging or binning are one solution for large datasets (e.g., tens of thousands of points).199 Divide and conquer was used here. To accomplish this, a first stage of unsupervised ML is performed with constrained kmeans clustering to produce 200 clusters with 500 to 1,000 points per cluster, suitable for making a piecewise surrogate. This level of cluster size is near the practical limit for conventional computing architectures of today with GPs. As noted, the use of GPs enables obtaining an estimated uncertainty for the predictions, and this can be compared to the actual error on untrained points as well as from statistical sampling between multiple choices of training sets.
These comparisons are important for trust, which is significant priority in safety-related applications.
There is research comparing various AI/ML methods for sparse dataset predictions, with NN showing good scaling and performance.200 NN may become more widespread in these applications if their overfitting and explainability can be improved. In this work, the piecewise GP surrogate model was used.
51 The following approach was used to create the piecewise GP surrogate model (pGP), with uncertainty quantification as a priority: (1) constrained kmeans clustering was performed to create regions for the piecewise surrogate. (2) For each cluster, GP regression was performed with 5-fold Monte Carlo Cross-validation with an 80 % / 20 % training/test split within each fold. The GP regressions were performed independently for the activities of each of the 10 elements. During the GP regression, the kernels evaluated were: Mat32, Mat52, radial basis function (RBF), Exponential, Cosine, and the kernel retained was whichever kernel achieved a regression coefficient of determination, r2, of greater than 0.97 first, or the kernel with the highest r2 (if no kernel achieved an r2 > 0.97). Within the 5-folds per elemental activity (for a given cluster), it was possible for different kernels to be chosen across the different folds.
The surrogate model then involves averaging predictions from this set of 5 GPs (for all 10 elements it is 50 GPs for a given cluster). The estimate of the final ML surrogate uncertainty of the prediction is taken as the greater of either the average one standard deviation uncertainty (1 uncertainty) returned by the 5 GPs (which is the composite mean GP predicted 1 uncertainty, <UGP>), or the 1 variability from the 5-fold cross-validation, UCV. This pair of uncertainties (one from the GP and one from statistical sampling) provides an ability to check the GPs ability to account for their epistemic uncertainty of predictions. The final surrogate uncertainty was taken as UF = max(<UGP>, UCV) for each elemental activity at each cluster.
Computational considerations were also important: for these applications, the methodology should linearly scalable and parallelizable, for both training and evaluation. On the order of 1000 GP points was needed for reasonable accuracy and to remain viable with conventional computing architectures (that is, with a < 5 GHz processor and < 40 GB). Accordingly, the full training data of 150000 points was divided piecewise by constrained kmeans clustering, with clusters constrained to 500-1000 points per cluster.
The total number of GPs within the piecewise surrogate model was thus 20,000 (from 200 clusters times 5 samplings per activity times 10 activities). This type of ML surrogate model does not compress the data:
the training data was 35 MB on disk, while the surrogate model is much larger, as the storage of GPs is known to scale at the order of O(n2) for n training points.201, 202 Total size on disk for the ML surrogate was
~80 GB when serialized, and >10 GB of memory if completely loaded into memory. However, the piecewise surrogate was utilized by cycling through the clusters (which could be parallelized) thus requiring < 10 GB of memory during predictions. It is appropriate to compare the time of prediction for each composition with the various methods. The CALPHAD calculations take 2-3 minutes per composition with a single processor. For the pGP model, the loading time was on the order of 10 seconds (per cluster),
and the evaluation for each predicted elemental activity at a given composition required, on average, 0.0052 s with a standard deviation of 0.0065 s, resulting in < 0.1 s per composition. A multi-linear regression OLS surrogate model was fitted to the same 150,000 points, for comparison. The training and testing for the OLS are orders of magnitude faster, with the model taking up << 1 MB of memory, but it is expected to be less accurate. Following this training and testing, the activities of 15,000 fresh points were used for validation. While the OLS model is the fastest, it is not accurate and not useful. In contrast, the pGP Model is the pGP model is sufficiently accurate (see Section 4.1) and on the order of 1000 times faster than the CALPHAD calculation while also providing estimated uncertainties.
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