Toward Using Surrogates to Accelerate Solution of Stochastic Electricity Grid Operations Problems.
Abstract not provided.
Safta, Cosmin↗
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Results for “Surrogates”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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30 CM HORIZONTAL DROP OF A SURROGATE 17X17 PWR FUEL ASSEMBLY.
Abstract not provided.
Kalinina, Elena Arkadievna↗
Molecular Dynamics Simulation of Fission Fragment Damage in Nuclear Fuel and Surrogate Material
We have performed classical molecular dynamics simulations of swift heavy ion damage, typical of fission fragments, in nuclear fuel (UO2) for energy deposition per unit length of 3.9 keV/nm. We did not observe amorphization. The damage mainly consisted of isolated point defects. Only about 1% of the displacements occur on the uranium sublattice. Oxygen Frenkel pairs are an order of magnitude more numerous than uranium Frenkel pairs in the primary damage state. In contrast, previous results show that the ratio of Frenkel pairs on the two sublattices is close to the stoichiometric ratio in ceria. These differences in the primary damage state may lead to differences in radiation response of UO2 and CeO2.
Molecular dynamics, radiation damage in materials,↗
Replicating HCCI-like autoignition behavior: What gasoline surrogate fidelity is needed?
This work seeks to characterize the fidelity needed in a gasoline surrogate with the intent to replicate the complex autoignition behavior exhibited within advanced combustion engines, and specifically Homogeneous Charge Compression Ignition (HCCI). A low-temperature gasoline combustion (LGTC) engine operating in HCCI mode and a rapid compression machine (RCM) are utilized to experimentally quantify fuel reactivity, through autoignition and preliminary heat release characteristics. Fuels considered include a research grade E10 U.S. gasoline (RD5-87), three multi-component surrogates (PACE-1, PACE-8, PACE-20), and a binary surrogate (PRF88.4). Each fuel was studied at lean/HCCI-like conditions covering a wide range of temperatures and pressures that are representative of naturally aspirated to high boost engine operation. Detailed chemical kinetic modeling is also undertaken using a recently updated gasoline surrogate kinetic model to simulate the RCM experiments and to provide chemical insight into surrogate-to-surrogate differences. The LGTC engine experiments demonstrate nearly identical reactivity between PACE-20 and RD5-87 across studied conditions, while faster phasing is seen for both PACE-1 and PACE-8 due to their stronger intermediate- and low-temperature heat release (ITHR/LTHR) at naturally aspirated and boosted conditions, respectively. The RCM experiments reveal typical low-temperature, negative temperature coefficient (NTC) and intermediate-temperature autoignition behaviors at all pressure conditions for RD5-87, which are qualitatively reproduced by all surrogates. Quantitative discrepancies in both autoignition and preliminary heat release are observed for all surrogates, while their ability to replicate RD5-87 autoignition behavior follows the order of PACE-20 > PACE-1 > PACE-8 > PRF88.4. Excellent mapping is obtained between the LGTC engine and the RCM, where the engine pressure-time trajectories can be characterized by the regimes represented by the RCM autoignition isopleths. The kinetic model performs commendably when simulating both autoignition and preliminary heat release of PACE-20, while typically overpredicting ignition delay times for PACE-1, PACE-8 and PRF88.4 at high-pressure and low-temperature/NTC conditions. Sensitivity and rate of production (ROP) analyses highlight surrogate-to-surrogate differences in the governing chemical kinetics where n-pentane initiates rapid OH branching at a faster rate and an earlier timing for PACE-20 than iso-pentane does for PACE-1 and PACE-8, making it computationally more reactive than the other surrogates. The current study highlights the need to include non-standardized properties, such as the lean/HCCI-like autoignition characteristics, in addition to ASTM properties (e.g., RON, MON) as metrics of fuel reactivity and targets to be matched when formulating high-fidelity surrogates that fully capture gasoline advanced combustion behavior such as HCCI-like autoignition.
42 ENGINEERING↗
Efficient surrogate modeling methods for large-scale Earth system models based on machine-learning techniques
Abstract. Improving predictive understanding of Earth system variability and change requires data–model integration. Efficient data–model integration for complex models requires surrogate modeling to reduce model evaluation time. However, building a surrogate of a large-scale Earth system model (ESM) with many output variables is computationally intensive because it involves a large number of expensive ESM simulations. In this effort, we propose an efficient surrogate method capable of using a few ESM runs to build an accurate and fast-to-evaluate surrogate system of model outputs over large spatial and temporal domains. We first use singular value decomposition to reduce the output dimensions and then use Bayesian optimization techniques to generate an accurate neural network surrogate model based on limited ESM simulation samples. Our machine-learning-based surrogate methods can build and evaluate a large surrogate system of many variables quickly. Thus, whenever the quantities of interest change, such as a different objective function, a new site, and a longer simulation time, we can simply extract the information of interest from the surrogate system without rebuilding new surrogates, which significantly reduces computational efforts. We apply the proposed method to a regional ecosystem model to approximate the relationship between eight model parameters and 42 660 carbon flux outputs. Results indicate that using only 20 model simulations, we can build an accurate surrogate system of the 42 660 variables, wherein the consistency between the surrogate prediction and actual model simulation is 0.93 and the mean squared error is 0.02. This highly accurate and fast-to-evaluate surrogate system will greatly enhance the computational efficiency of data–model integration to improve predictions and advance our understanding of the Earth system.
58 GEOSCIENCES↗
Efficient Loads Surrogates for Waked Turbines in an Array
Accurately and efficiently predicting wind turbine structural loading is a crucial step in wind farm design. Without considering structural loading, wind farm optimization could negatively impact turbine fatigue and ultimate loads, especially for waked and partially waked turbines, which could result in higher maintenance costs and reduced turbine lifetime. However, predicting turbine loads throughout an array is a costly step, as these quantities require time-accurate results across long time histories, which is often intractable for large array optimization. Therefore, surrogate models that link array spacing to load outputs are often used, but the surrogates are then unique to the inflow conditions and array configurations in the training library. This work develops surrogate models for many wind turbine load outputs based solely on rotor plane velocity measurements, with no required input about array configuration or freestream inflow parameters. Surrogate models were constructed for many turbine quantities of interest (QoI), considering mean, standard deviation, ultimate, and fatigue loads. In general, most QoI statistics were accurately captured, as measured by predicted vs. actual correlation coefficient, confirming the suitability of the approach. Temporal mean values of the QoI required only temporal mean measurements of the rotor plane inflow velocity. However, accurate prediction of temporal standard deviation, ultimate, and fatigue values of QoI also required temporal standard deviations of the rotor plane velocity field. Poor surrogate performance was observed when too many correlated inputs were used, such as multiple velocity components. If the fewest inflow parameters are used to construct the surrogates, the average correlation coefficient value for all output QoI statistics is 0.89. Surrogates for standard deviations and damage equivalent loads (DELs) of turbine QoIs generally had lower accuracy and tower-base and shaft load channels posed the most difficult to capture accurately. The results suggest that these surrogates could be easily paired with analytic wake models, which are frequently used for pre-construction wind farm array optimization, to account for turbine loading in addition to power production. By including the optimal inflow conditions, the surrogate accuracy can improve to an average correlation coefficient value for all output QoI statistics of 0.92. This work has established the ability to build accurate surrogates for mean, standard deviation, ultimate load, and DEL turbine QoI values based on the rotor plane inflow velocity, and identified which inflow conditions lead to greater surrogate accuracy.
17 WIND ENERGY↗
Light Water Reactor Sustainability Program Status of Adaptive Surrogates within the RAVEN framework
The RAVEN code has been under development at the Idaho National Laboratory since 2012. Its main goal is to create a multi-purpose platform for the deploying of all the capabilities needed for Probabilistic Risk Assessment, uncertainty quantification, data mining analysis and optimization studies. RAVEN has demonstrated a good level of maturity in terms of state-of-art and advanced analysis methodologies. Indeed, in the past year the RAVEN code has been released as open-source project and is available to download (raven.inl.gov) free of charge. The main subject of this report is to show the activities that have been recently accomplished with respect the adaptive surrogate modeling: Implementation of the metric system for the assessment of surrogate model validity; Implementation of cross-validation techniques for the global assessment of surrogate models; Implementation of an automatic infrastructure for switching between a high-fidelity model (e.g. physical code) and a surrogate or set of surrogates; Implementation of a scheme for automatic optimization of surrogate model parameters. The aim of this document is to report the status of the development of methods, within the RAVEN framework, to assess the validity of the predictive capabilities of surrogate models. Indeed, after the construction of a surrogate tight to a certain physical model, it is crucial to assess the goodness of its representation, in order to be confident with its prediction. A cross-validation technique has been employed. This report will highlight the implementation details and proof its correct implementation by an application example. The implementation of a way to assess the predictive capabilities of a surrogate allowed the construction of a RAVEN new entity named “HybridModel” that is aimed to switch between an high-fidelity code and its linked surrogate. In addition, leveraging the optimization algorithms that have been implemented in RAVEN last fiscal year, a new scheme has been developed for automatic tuning of the surrogates’ parameters (e.g. penalty factors in support vector machines).
97 MATHEMATICS AND COMPUTING↗
Adaptive experimental design for multi-fidelity surrogate modeling of multi-disciplinary systems
We present an adaptive algorithm for constructing surrogate models of multi-disciplinary systems composed of a set of coupled components. With this goal we introduce “coupling” variables with a priori unknown distributions that allow surrogates of each component to be built independently. Once built, the surrogates of the components are combined to form an integrated-surrogate that can be used to predict system-level quantities of interest at a fraction of the cost of the original model. The error in the integrated-surrogate is greedily minimized using an experimental design procedure that allocates the amount of training data, used to construct each component-surrogate, based on the contribution of those surrogates to the error of the integrated-surrogate. Here, the multi-fidelity procedure presented is a generalization of multi-index stochastic collocation that can leverage ensembles of models of varying cost and accuracy, for one or more components, to reduce the computational cost of constructing the integrated-surrogate. Extensive numerical results demonstrate that, for a fixed computational budget, our algorithm is able to produce surrogates that are orders of magnitude more accurate than methods that treat the integrated system as a black-box.
42 ENGINEERING↗
Soot formation and precursor chemistry in Counterflow flames of aviation fuel surrogates
To meet market demands, the aviation sector is interested in utilizing drop-in Synthetic Aviation Turbine Fuels (SATF), either as neat fuels or in blends with conventional Jet A. SATF currently approved in standard specifications may have lower aromatic content with significant fractions of normal, branched, and cyclo-alkanes. Fundamental studies on soot formation from aviation fuels (Jet A, SATF) and their surrogate components are essential to understand how fuel composition influences soot and soot precursor formation. Here, this study reports new measurements of polycyclic aromatic hydrocarbons (PAH) and soot in counterflow diffusion flames (CDFs) of aviation fuel surrogates. Both intrusive and non-intrusive diagnostics are employed to determine the profiles of temperature, gas phase species, PAHs (up to C16), and soot volume fraction (SVF) in CDFs of iso-octane and surrogate mixtures. These measurements shed light on the transition of soot precursors to primary soot particles. In addition to serving as a common surrogate component in Jet A surrogate mixtures, iso-octane is a template species for larger, less volatile branched alkanes found in SATF mixtures. The newly developed Lawrence Livermore National Laboratory (LLNL) PAH and soot model successfully captures temperature, precursor species, and SVF profiles for the mixtures and conditions discussed in this work. Finally, a high-fidelity surrogate for Jet A is proposed that matches targeted physical and chemical properties well, while leveraging the wide range of candidate fuel molecules available in the LLNL detailed chemical model. The proposed surrogate formulation is validated against newly acquired measurements of the surrogate and literature measurements of Jet A. These new experiments and simulations provide critical insights into the PAH and soot formation from aviation fuels. Reaction pathways which require further investigation are highlighted, such that future work may bridge the remaining quantitative gaps in predicting soot formation from aviation fuel surrogates and surrogate components.
Aviation fuels↗
FAIR Surrogate Benchmarks Supporting AI and Simulation Research (Final Report)
Computational Science is being revolutionized by integrating AI and simulation and, in particular, by deep learning surrogate models that can replace all or part of traditional large‐scale HPC computations. Such surrogates can achieve remarkable performance improvements, as much as several orders of magnitude, and save both compute time and energy. The Surrogate Benchmark Initiative (SBI) project creates a community repository and FAIR (Findable, Accessible, Interoperable, and Reusable) data ecosystem for HPC application surrogate benchmarks. The SBI team comes from Argonne National Laboratory (ANL), Indiana University (IU), Rutgers University, the University of Tennessee, Knoxville (UTK), and the University of Virginia (UVA). SBI repositories include data, code, and all relevant collateral artifacts that the science and engineering community need to use and reuse these data sets and surrogates. SBI repositories generate active research from both the participants in SBI and the broad community of AI and domain scientists. This project develops surrogates that use several different neural nets to learn and quickly infer the results of simulations and data systems and captures them as surrogate benchmarks with a rich set of metadata covering: Data; Model; Metrics specification; Machine specification; and Science, Speed, and Power Results. We research FAIR metadata for these benchmarks. We develop application surrogate examples as benchmarks across many fields (ANL, UTK, IU, UVA). We also study non-Surrogate benchmarks that have many common features and similar issues as regards FAIRness. We work with MLCommons (UVA, UTK), which is a major machine learning benchmarking activity where we get metadata ontologies, software, and benchmarks, Benchmarks have datasets, models, and metadata and they need a technical framework developed by UTK and Rutgers and deployed by UVA. We study features of Surrogates including performance, training set size, and uncertainty quantification (Rutgers, UVA and IU).
97 MATHEMATICS AND COMPUTING↗
Bayesian model updating with finite element vs surrogate models: Application to a miter gate structural system
Bayesian finite element (FE) model updating using direct model evaluations of large-scale high-fidelity FE models is extremely computationally expensive. Surrogate models can be used as fast emulators of FE models to accelerate the model calibration process. The physics/mechanics-based FE models are still the underpinning behind the surrogate models. Here, this paper evaluates the loss in accuracy and the gain in computational time while performing Bayesian model updating by using surrogate model evaluations compared to using direct FE model evaluations. This evaluation is crucial before entirely relying on surrogate models in model updating for structural health monitoring (SHM) and damage prognosis (DP) purposes. This paper also demonstrates Bayesian updating and surrogate model construction of large-scale high-fidelity FE models of infrastructure systems. In this regard, the miter gate structural system is considered as the testbed structure. Three predominant damage modes (loss of contact between gate and wall, loss of thickness due to corrosion, and loss of tension in the diagonal rods) are considered for model updating purposes. Bayesian model updating is performed using direct FE evaluations by leveraging parallel computing. Two types of surrogates, namely polynomial chaos expansion (PCE) and Gaussian process regression (GPR), are developed for the miter gate. Model updating is performed again using the trained surrogate models, and the updating results are compared with their counterparts obtained using the direct FE evaluation results. The posterior distribution of the FE model parameters obtained using the trained surrogates are sufficiently accurate with respect to the posterior obtained utilizing the direct FE evaluations. In addition, an approximate 4-fold decrease in the computational time was observed when using surrogate model evaluations instead of direct FE evaluations for model updating.
42 ENGINEERING↗
Cross sections for neutron-induced reactions from surrogate data: Reexamining the Weisskopf-Ewing approximation for ( n , n ' ) and ( n , 2 n ) reactions
Background: Modeling nuclear reaction networks for nuclear science applications and for simulations of astrophysical environments relies on cross section data for a vast number of reactions, many of which have never been measured. Cross sections for neutron-induced reactions on unstable nuclei are particularly scarce, since they are the most difficult to measure. Consequently, we must rely on theoretical predictions or indirect measurements to obtain the requisite reaction data. For compound nuclear reactions, the surrogate reaction method can be used to determine many cross sections of interest. Purpose: Earlier work has demonstrated that cross sections for neutron-induced fission and radiative neutron capture can be determined from a combination of surrogate reaction data and theory. For the fission case, it was shown that the Weisskopf-Ewing approximation, which significantly simplifies the implementation of the surrogate method, can be employed. Capture cross sections cannot be obtained, and require a detailed description of the surrogate reaction process. Here, we examine the validity of the Weisskopf-Ewing approximation for determining unknown (n, n') and (n, 2n) cross sections from surrogate data. Methods: Using statistical reaction calculations with realistic parametrizations, we investigate first whether the assumptions underlying the Weisskopf-Ewing approximation are valid for (n, n') and (n, 2n) reactions on representative target nuclei. We then produce simulated surrogate reaction data and assess the impact of applying the Weisskopf-Ewing approximation when extracting (n, n') and (n, 2n) cross sections in situations where the approximation is not strictly justified. Results: We find that peak cross sections can be estimated using the Weisskopf-Ewing approximation, but the shape of the (n, n') and (n, 2n) cross sections, especially for low neutron energies, cannot be reliably determined without accounting for the angular-momentum differences between the neutron-induced and surrogate reaction. Conclusions: To obtain reliable (n, n') and (n, 2n) cross sections from surrogate reaction data, a detailed description of the surrogate reaction mechanisms is required. To do so for the compound-nucleus energies and decay channels relevant to these reactions, it becomes necessary to extend current modeling capabilities.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Shape Optimization by Bayesian-Validated Computer-Simulation Surrogates
A nonparametric-validated, surrogate approach to optimization has been applied to the computational optimization of eddy-promoter heat exchangers and to the experimental optimization of a multielement airfoil. In addition to the baseline surrogate framework, a surrogate-Pareto framework has been applied to the two-criteria, eddy-promoter design problem. The Pareto analysis improves the predictability of the surrogate results, preserves generality, and provides a means to rapidly determine design trade-offs. Significant contributions have been made in the geometric description used for the eddy-promoter inclusions as well as to the surrogate framework itself. A level-set based, geometric description has been developed to define the shape of the eddy-promoter inclusions. The level-set technique allows for topology changes (from single-body,eddy-promoter configurations to two-body configurations) without requiring any additional logic. The continuity of the output responses for input variations that cross the boundary between topologies has been demonstrated. Input-output continuity is required for the straightforward application of surrogate techniques in which simplified, interpolative models are fitted through a construction set of data. The surrogate framework developed previously has been extended in a number of ways. First, the formulation for a general, two-output, two-performance metric problem is presented. Surrogates are constructed and validated for the outputs. The performance metrics can be functions of both outputs, as well as explicitly of the inputs, and serve to characterize the design preferences. By segregating the outputs and the performance metrics, an additional level of flexibility is provided to the designer. The validated outputs can be used in future design studies and the error estimates provided by the output validation step still apply, and require no additional appeals to the expensive analysis. Second, a candidate-based a posteriori error analysis capability has been developed which provides probabilistic error estimates on the true performance for a design randomly selected near the surrogate-predicted optimal design.
Patera, Anthony T.↗
Iterative sampling of expensive simulations for faster deep surrogate training
Deep neural network (DNN) surrogates of expensive physics simulations are enabling a rapid change in the way that common experimental design and analysis tasks are approached. Surrogate models allow simulations to be performed in parallel and separately from downstream tasks, thereby enabling analyses that would be impossible with the simulation in-the-loop; surrogates based on DNNs can effectively emulate diverse non-scalar data of the types collected in fusion and laboratory-astrophysics experiments. The challenge is in training the surrogate model, for which large ensembles of physics simulations must be run, preferably without wasting computational effort on uninteresting simulations. Here, in this paper, we present an iterative sampling scheme that can preferentially propose simulations in interesting regions of parameter space without neglecting unexplored regions, allowing high-quality and wide-ranging surrogate models to be trained using 2–3 times fewer simulations compare to space-filling designs. Our approach uses an explicit importance function defined on the simulation output space, balanced against a measure of simulation density which serves as a proxy for surrogate accuracy. It is easy to implement and can be tuned to find interesting simulations early in the study, allowing surrogates to be trained quickly and refined as new simulations become available; this represents an important step towards the routine generation of deep surrogate models quickly enough to be truly relevant to experimental work.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Efficient Surrogate Model Development: Impact of Sample Size and Underlying Model Dimensions
This study compares eight surrogate-model construction approaches using computational experiments. Here, the construction approaches considered include: Artificial Neural Networks (ANNs), Automated Learning of Algebraic Models using Optimization (ALAMO), Radial Basis Networks (RBNs), Extreme Learning Machines (ELMs), Gaussian Progress Regression (GPR), Random Forests (RFs), Support Vector Regression (SVR), and Multivariate Adaptive Regression Splines. Each approach is used to construct surrogate models for predicting the outputs of thirty-four test functions. The data used to construct surrogate models were generated using Latin Hypercube Sampling (LHS), Halton and Sobol sampling methods. We studied the impact of sampling method, sample size, and challenge function characteristics on the accuracy of surrogate model predictions. Our results revealed that for smaller sample sizes, on average, surrogate models trained using the data points generated by Sobol sequences provided the best estimation for all surrogate model construction approaches. As the sample size increased, the impact of sampling methods diminished. The comparison of surrogate-modeling construction approaches at large sample sizes revealed that surrogate models trained using ANN, ALAMO, and ELM yielded smaller root mean squared error and higher adjusted R-squared values than the models trained using the rest of the approaches.
42 ENGINEERING↗
A framework for strategic discovery of credible neural network surrogate models under uncertainty
The widespread integration of deep neural networks in developing data-driven surrogate models for high-fidelity simulations of complex physical systems highlights the critical necessity for robust uncertainty quantification techniques and credibility assessment methodologies, ensuring the reliable deployment of surrogate models in consequential decision-making. Here, this study presents the Occam Plausibility Algorithm for surrogate models (OPAL-surrogate), providing a systematic framework to uncover predictive neural network-based surrogate models within the large space of potential models, including various neural network classes and choices of architecture and hyperparameters. The framework is grounded in hierarchical Bayesian inferences and employs model validation tests to evaluate the credibility and prediction reliability of the surrogate models under uncertainty. Leveraging these principles, OPAL-surrogate introduces a systematic and efficient strategy for balancing the trade-off between model complexity, accuracy, and prediction uncertainty. The effectiveness of OPAL-surrogate is demonstrated through two modeling problems, including the deformation of porous materials for building insulation and turbulent combustion flow for ablation of solid fuels within hybrid rocket motors.
42 ENGINEERING↗
High-pressure ignition delay time measurements of a four-component gasoline surrogate and its high-level blends with ethanol and methyl acetate
Ignition delay times of a gasoline surrogate (iso-octane/toluene/n-heptane/1-hexene at 55/25/15/5% by liquid volume, developed in McCormick et al. 2017) with high-level bioblendstock-gasoline surrogate blends (50% and 85% biofuel by liquid volume of each ethanol and methyl acetate) were collected behind reflected shock waves. Post-reflected-shock temperatures ranged from 968 to 1361 K at pressures of about 4, 10, 25, and 50 atm. Data were collected for real fuel–air mixtures at fuel lean and stoichiometric conditions (φ = 0.5, 1.0) with focus on the more-practical, fuel-lean conditions. Ignition delay times were measured from OH* chemiluminescence around 307 nm using an endwall diagnostic. The RON and MON of the surrogate were 90.3 and 84.7, respectively. Ethanol was chosen due to its wide use in flex-fuel vehicles and availability, while also providing an increased octane rating. Methyl acetate was chosen for its especially high octane rating to investigate the effect of an extreme case. To validate the gasoline surrogate, the data are compared to real gasoline (RD387) and several gasoline surrogate experiments from the literature. Using the wide range of pressures studied, the gasoline surrogate’s pressure dependence was quantified to account for test-to-test variations using regression analysis. Similarly, a global correlation for gasoline and its surrogates was developed using all available data from the literature. Two modern chemical kinetics models targeting gasoline and its surrogates are compared to the ignition delay time measurements. These new high-pressure, high-bioblendstock concentration tests provide required chemical kinetic data for optimizing fuel and engine design.
42 ENGINEERING↗
Surrogate modeling for efficiently, accurately and conservatively estimating measures of risk
We present a surrogate modeling framework for conservatively estimating measures of risk from limited realizations of an expensive physical experiment or computational simulation. Risk measures combine objective probabilities with the subjective values of a decision maker to quantify anticipated outcomes. Given a set of samples, we construct a surrogate model that produces estimates of risk measures that are always greater than their empirical approximations obtained from the training data. These surrogate models limit over-confidence in reliability and safety assessments and produce estimates of risk measures that converge much faster to the true value than purely sample-based estimates. We first detail the construction of conservative surrogate models that can be tailored to a stakeholder’s risk preferences and then present an approach, based on stochastic orders, for constructing surrogate models that are conservative with respect to families of risk measures. Our surrogate models include biases that permit them to conservatively estimate the target risk measures. We provide theoretical results that show that these biases decay at the same rate as the L 2 error in the surrogate model. Numerical demonstrations confirm that risk-adapted surrogate models do indeed overestimate the target risk measures while converging at the expected rate.
42 ENGINEERING↗