Machine learned chemical potentials have shown great promise as alternatives to conventional computational chemistry methods to represent the potential energy of a given atomic or molecular system as a function of its geometry. However, such potentials are only as good as the data they are trained on, and building a comprehensive training set can be a costly process. Therefore, it is important to extract as much information from training data as possible without further increasing the computational cost. One way to accomplish this is by training on molecular forces in addition to energies. This allows for three additional labels per atom within the molecule. Here we develop a neural network potential energy surface for studying a hydrogen transfer reaction between two conformers of C5H5. We show that, for a much smaller training set, force training can greatly improve the accuracy of the model compared to only training to energies. We also demonstrate the importance of choosing the proper force to energy weight ratio for the loss function to minimize the model test error.

This paper analyzes the approximate control variate (ACV) approach to multifidelity uncertainty quantification in the case where weighted estimators are combined to form the components of the ACV. The weighted estimators enable one to precisely group models that share input samples to achieve improved variance reduction. We demonstrate that this viewpoint yields a generalized linear estimator that can assign any weight to any sample. This generalization shows that other linear estimators in the literature, particularly the multilevel best linear unbiased estimator (ML-BLUE) of Schaden and Ullman in 2020, becomes a specific version of the ACV estimator of Gorodetsky, Geraci, Jakeman, and Eldred, 2020. Moreover, this connection enables numerous extensions and insights. For example, we empirically show that having non-independent groups can yield better variance reduction compared to the independent groups used by ML-BLUE. Furthermore, we show that such grouped estimators can use arbitrary weighted estimators, not just the simple Monte Carlo estimators used in ML-BLUE. Furthermore, the analysis enables the derivation of ML-BLUE directly from a variance reduction perspective, rather than a regression perspective.

Optimization is a key tool for scientific and engineering applications; however, in the presence of models affected by uncertainty, the optimization formulation needs to be extended to consider statistics of the quantity of interest. Optimization under uncertainty (OUU) deals with this endeavor and requires uncertainty quantification analyses at several design locations; i.e., its overall computational cost is proportional to the cost of performing a forward uncertainty analysis at each design location. An OUU workflow has two main components: an inner loop strategy for the computation of statistics of the quantity of interest, and an outer loop optimization strategy tasked with finding the optimal design, given a merit function based on the inner loop statistics. In this work, we propose to alleviate the cost of the inner loop uncertainty analysis by leveraging the so-called multilevel Monte Carlo (MLMC) method, which is able to allocate resources over multiple models with varying accuracy and cost. The resource allocation problem in MLMC is formulated by minimizing the computational cost given a target variance for the estimator. We consider MLMC estimators for statistics usually employed in OUU workflows and solve the corresponding allocation problem. For the outer loop, we consider a derivative -free optimization strategy implemented in the SNOWPAC library; our novel strategy is implemented and released in the Dakota software toolkit. We discuss several numerical test cases to showcase the features and performance of our approach with respect to its Monte Carlo single fidelity counterpart.

In order to make design decisions, engineers may seek to identify regions of the design domain that are acceptable in a computationally efficient manner. A design is typically considered acceptable if its reliability with respect to parametric uncertainty exceeds the designer’s desired level of confidence. Despite major advancements in reliability estimation and in design classification via decision boundary estimation, the current literature still lacks a design classification strategy that incorporates parametric uncertainty and desired design confidence. To address this gap, this works offers a novel interpretation of the acceptance region by defining the decision boundary as the hypersurface which isolates the designs that exceed a user-defined level of confidence given parametric uncertainty. This work addresses the construction of this novel decision boundary using computationally efficient algorithms that were developed for reliability analysis and decision boundary estimation. The proposed approach is verified on two physical examples from structural and thermal analysis using Support Vector Machines and Efficient Global Optimization-based contour estimation.

Multifidelity uncertainty quantification (MF UQ) sampling approaches have been shown to significantly reduce the variance of statistical estimators while preserving the bias of the highest-fidelity model, provided that the low-fidelity models are well correlated. However, maintaining a high level of correlation can be challenging, especially when models depend on different input uncertain parameters, which drastically reduces the correlation. Existing MF UQ approaches do not adequately address this issue. In this work, we propose a new sampling strategy that exploits a shared space to improve the correlation among models with dissimilar parametrization. We achieve this by transforming the original coordinates onto an auxiliary manifold using the adaptive basis (AB) method~\cite{Tipireddy2014}. The AB method has two main benefits: (1) it provides an effective tool to identify the low-dimensional manifold on which each model can be represented, and (2) it enables easy transformation of polynomial chaos representations from high- to low-dimensional spaces. This latter feature is used to identify a shared manifold among models without requiring additional evaluations. We present two algorithmic flavors of the new estimator to cover different analysis scenarios, including those with legacy and non-legacy high-fidelity data. We provide numerical results for analytical examples, a direct field acoustic test, and a finite element model of a nuclear fuel assembly. For all examples, we compare the proposed strategy against both single-fidelity and MF estimators based on the original model parametrization.

We analyze the regression accuracy of convolutional neural networks assembled from encoders, decoders and skip connections and trained with multifidelity data. Besides requiring significantly less trainable parameters than equivalent fully connected networks, encoder, decoder, encoder-decoder or decoder-encoder architectures can learn the mapping between inputs to outputs of arbitrary dimensionality. We demonstrate their accuracy when trained on a few high-fidelity and many low-fidelity data generated from models ranging from one-dimensional functions to Poisson equation solvers in two-dimensions. We finally discuss a number of implementation choices that improve the reliability of the uncertainty estimates generated by Monte Carlo DropBlocks, and compare uncertainty estimates among low-, high- and multifidelity approaches.

View Video Presentation: https://doi.org/10.2514/6.2023-1481.vid Multi-model Monte Carlo methods are efficient strategies to perform forward uncertainty quantification studies in entry, descent, and landing (EDL) applications. These multi-model methods are based on the classical Monte Carlo estimator, but fuse predictions from several low-fidelity models to obtain estimators with greater precision given a prescribed computational budget. The effectiveness of these approaches relies on the magnitudes of correlations between the low-fidelity models and the high-fidelity model, as well as the relative computational costs of all models. Identifying and exploiting the best trade-off between correlation and cost, which ultimately depends on the selection of hyperparameters in the low-fidelity models, is a task often performed by hand or simply inspired by the deterministic understanding available for a specific application. This work extends a preliminary effort, which demonstrated the importance of automatic and optimal tuning of hyperparameters to obtain efficient multi-model estimators. New optimization strategies are compared to gauge their feasibility and limitations for realistic applications, e.g., by accounting for their computational demand. The Sandia National Laboratories software Dakota, which supports a variety of multi-model estimators and optimization algorithms, is employed in all numerical examples for both analytical tests and EDL calculations. A case study in the Adaptable, Deployable Entry and Placement Technology (ADEPT) project trajectory simulations demonstrates the practical advantages of the proposed framework.

Surrogate construction is an essential component for all non-deterministic analyses in science and engineering. The efficient construction of easy and cheaper-to-run alternatives to a computationally expensive code paves the way for outer loop workflows for forward and inverse uncertainty quantification and optimization. Unfortunately, the accurate construction of a surrogate still remains a task that often requires a prohibitive number of computations, making the approach unattainable for large-scale and high-fidelity applications. Multifidelity approaches offer the possibility to lower the computational expense requirement on the high-fidelity code by fusing data from additional sources. In this context, we have demonstrated that multifidelity Bayesian Networks (MFNets) can efficiently fuse information derived from models with an underlying complex dependency structure. In this contribution, we expand on our previous work by adopting a basis adaptation procedure for the selection of the linear model representing each data source. Our numerical results demonstrate that this procedure is computationally advantageous because it can maximize the use of limited data to learn and exploit the important structures shared among models. Two examples are considered to demonstrate the benefits of the proposed approach: an analytical problem and a nuclear fuel finite element assembly. From these two applications, a lower dependency of MFnets on the model graph has been also observed.

This paper focuses on the development of multifidelity modeling approaches using neural network surrogates, where training data arising from multiple model forms and resolutions are integrated to predict high-fidelity response quantities of interest at lower cost. We focus on the context of quantum chemistry and the integration of information from multiple levels of theory. Important foundations include the use of symmetry function-based atomic energy vector constructions as feature vectors for representing structures across families of molecules and single-fidelity neural network training capabilities that learn the relationships needed to map feature vectors to potential energy predictions. These foundations are embedded within several multifidelity topologies that decompose the highfidelity mapping into model-based components, including sequential formulations that admit a general nonlinear mapping across fidelities and discrepancy-based formulations that presume an additive decomposition. Methodologies are first explored and demonstrated on a pair of simple analytical test problems and then deployed for potential energy prediction for C5H5 using B2PLYP-D3/6-311++G(d,p) for high-fidelity simulation data and Hartree-Fock 6-31G for low-fidelity data. For the common case of limited access to high-fidelity data, our computational results demonstrate that multifidelity neural network potential energy surface constructions achieve roughly an order of magnitude improvement, either in terms of test error reduction for equivalent total simulation cost or reduction in total cost for equivalent error.

technique has previously been developed by Cooper and Larsen to reduce variance in global problems by distributing particles more evenly throughout the spatial domain. This hybrid method uses an approximate deterministic estimate of the forward scalar flux distribution to automatically generate weight windows for the Monte Carlo transport simulation, avoiding the necessity for the code user to specify the weight window parameters. In a binary stochastic medium, the material properties at a given spatial location are known only statistically. The most common approach to solving particle transport problems involving binary stochastic media is to use the atomic mix (AM) approximation in which the transport problem is solved using ensemble-averaged material properties. The most ubiquitous deterministic model developed specifically for solving binary stochastic media transport problems is the Levermore-Pomraning (L-P) model. Zimmerman and Adams proposed a Monte Carlo algorithm (Algorithm A) that solves the Levermore-Pomraning equations and another Monte Carlo algorithm (Algorithm B) that is more accurate as a result of improved local material realization modeling. Recent benchmark studies have shown that Algorithm B is often significantly more accurate than Algorithm A (and therefore the L-P model) for deep penetration problems such as examined in this paper. In this research, we investigate the application of a variant of the hybrid Monte Carlo-deterministic method proposed by Cooper and Larsen to global deep penetration problems involving binary stochastic media. To our knowledge, hybrid Monte Carlo-deterministic methods have not previously been applied to problems involving a stochastic medium. We investigate two approaches for computing the approximate deterministic estimate of the forward scalar flux distribution used to automatically generate the weight windows. The first approach uses the atomic mix approximation to the binary stochastic medium transport problem and a low-order discrete ordinates angular approximation. The second approach uses the Levermore-Pomraning model for the binary stochastic medium transport problem and a low-order discrete ordinates angular approximation. In both cases, we use Monte Carlo Algorithm B with weight windows automatically generated from the approximate forward scalar flux distribution to obtain the solution of the transport problem.

In this work we investigate the performance of a novel derivative-free optimization under uncertainty (OUU) algorithm based on Multilevel Monte Carlo (MLMC) for wind power plants. Our optimization algorithm combines the flexibility of derivative-free algorithms, which can be easily applied to black-box codes