Multifidelity uncertainty propagation combines the efficiency of low-fidelity models with the accuracy of a high-fidelity model to construct statistical estimators of quantities of interest. It is well known that the effectiveness of such methods depends crucially on the relative correlations and computational costs of the available computational models. However, the question of how to automatically tune low-fidelity models to maximize performance remains an open area of research. This work investigates automated model tuning, which optimizes model hyperparameters to minimize estimator variance within a target computational budget. Focusing on multifidelity trajectory simulation estimators, the cost-versus-precision tradeoff enabled by this approach is demonstrated in a practical, online setting where upfront tuning costs cannot be amortized. Using a real-world entry, descent, and landing example, it is shown that automated model tuning largely outperforms hand-tuned models even when the overall computational budget is relatively low. Furthermore, for scenarios where the computational budget is large, model tuning solutions can approach the best-case multifidelity estimator performance where optimal model hyperparameters are known a priori. Recommendations for applying model tuning in practice are provided and avenues for enabling adoption of such approaches for budget-constrained problems are highlighted.

The ability to accurately model random fields plays a critical role in science and engineering for problems involving uncertain, spatially-varying quantities such as heterogeneous material properties and turbulent flows. Deep generative models offer a powerful tool for sampling high- or infinite-dimensional uncertainties like random fields, but their reliance on large, dense training datasets limits their applicability in contexts where sufficient data is difficult or expensive to obtain. In this work, we propose a latent-space approach to generative modeling of random fields that incorporates domain knowledge to supplement limited training data. A constraint-aware variational autoencoder (VAE) with a function decoder is first used to learn compact latent representations of continuous functions that adhere to known physical or statistical constraints, even when training data is sparse or indirect. Generative modeling is then performed in the learned latent space, decoupling constraint enforcement from the sampling process. This decoupling enables expressive multi-step generative methods to be deployed in data-limited settings where existing constrained multi-step approaches are not directly applicable. The richer latent distributions captured by the generative model also overcome limitations of standard VAEs, which rely on simple parametric priors and struggle to represent complex, multimodal, or heavy-tailed distributions over functions. Efficacy is demonstrated on two challenging applications: wind velocity field reconstruction from sparse sensors and material property inference from indirect measurements. Results show the effectiveness of incorporating domain knowledge constraints for data-limited problems and the improved sample quality and robustness of the latent generative modeling approach versus directly sampling a constrained VAE.

Stochastic optimization of engineering systems is often infeasible due to repeated evaluations of a computationally expensive, high-fidelity simulation. Bi-fidelity methods mitigate this challenge by leveraging a cheaper, approximate model to accelerate convergence. Most existing bi-fidelity approaches, however, exploit either design-space curvature or random-space correlation, not both. We present BISTRO - a BI-fidelity Stochastic Trust-Region Optimizer for unconstrained optimization under uncertainty through a stochastic approximation procedure. This approach exploits the curvature information of a low-fidelity objective function to converge within a basin of a local minimum of the high-fidelity model where low-fidelity curvature information is no longer valuable. The method then switches to a variance-reduced stochastic gradient descent procedure. We provide convergence guarantees in expectation under certain regularity assumptions and ensure the best-case 𝒪(1/n) convergence rate for stochastic optimization. On benchmark problems and a 20-dimensional space shuttle reentry case, BISTRO converges faster than adaptive sampling and variance reduction procedures and cuts computational expense by up to 29x.

We provide a collection of results on covariance expressions between Monte Carlo--based multioutput mean, variance, and Sobol main effect variance estimators from an ensemble of models. These covariances can be used within multifidelity uncertainty quantification strategies that seek to reduce the estimator variance of high-fidelity Monte Carlo estimators with an ensemble of low-fidelity models. Such covariance expressions are required within approaches such as the approximate control variate and multilevel best linear unbiased estimator. While the literature provides these expressions for some single-output cases such as mean and variance, our results are relevant to both multiple function outputs and multiple statistics across any sampling strategy. Following the description of these results, we use them within an approximate control variate scheme to show that leveraging multiple outputs can dramatically reduce estimator variance compared to single-output approaches. Synthetic examples are used to highlight the effects of optimal sample allocation and pilot sample estimation. A flight-trajectory simulation of entry, descent, and landing is used to demonstrate multioutput estimation in practical applications.

Lifelong development allows animals and machines to adapt to changes in the environment as well as in their own systems, such as wear and tear in sensors and actuators. An important use case of such adaptation is industrial odor-sensing. Metal-oxide-based sensors can be used to detect gaseous compounds in the air; however, the gases interact with the sensors, causing their responses to change over time in a process called sensor drift. Sensor drift is irreversible and requires frequent recalibration with additional data. This paper demonstrates that an adaptive system that represents the drift as context for the skill of odor sensing achieves the same goal automatically. After it is trained on the history of changes, a neural network predicts future contexts, allowing the context+skill sensing system to adapt to sensor drift. Evaluated on an industrial dataset of gas-sensor drift, the approach performed better than standard drift-naive and ensembling methods. In this way, the context+skill system emulates the natural ability of animal olfaction systems to adapt to a changing world, and demonstrates how it can be effective in real-world applications.

In reliability analysis, methods used to estimate failure probability are often limited by the costs associated with model evaluations. Many of these methods, such as multifidelity importance sampling (MFIS), rely upon a computationally efficient surrogate model like a Gaussian process (GP) to quickly generate predictions. The quality of the GP fit, particularly in the vicinity of the failure region(s), is instrumental in supplying accurately predicted failures for such strategies. We introduce an entropy-based GP adaptive design that, when paired with MFIS, provides more accurate failure probability estimates and with higher confidence. We show that our greedy data acquisition strategy better identifies multiple failure regions compared to existing contour-finding schemes. We then extend the method to batch selection, without sacrificing accuracy. Illustrative examples are provided on benchmark data as well as an application to an impact damage simulator for National Aeronautics and Space Administration (NASA) spacesuits.

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.

We provide a collection of results on covariance expressions between Monte Carlo based multi-output mean, variance, and Sobol main effect variance estimators from an ensemble of models. These covariances can be used within multi-fidelity uncertainty quantification strategies that seek to reduce the estimator variance of high-fidelity Monte Carlo estimators with an ensemble of low-fidelity models. Such covariance expressions are required within approaches like the approximate control variate and multi-level best linear unbiased estimator. While the literature provides these expressions for some single-output cases such as mean and variance, our results are relevant to both multiple function outputs and multiple statistics across any sampling strategy. Following the description of these results, we use them within an approximate control variate scheme to show that leveraging multiple outputs can dramatically reduce estimator variance compared to single-output approaches. Synthetic examples are used to highlight the effects of optimal sample allocation and pilot sample estimation. A flight-trajectory simulation of entry, descent, and landing is used to demonstrate multi-output estimation in practical applications.

View Video Presentation: https://doi.org/10.2514/6.2022-0632.vid Reliability computation for transonic aeroelastic systems is a challenging task, given the complex physical mechanisms involved, and broader challenges associated with computing small probabilities of failure/flutter. This paper demonstrates the use of four different reliability estimation tools, with a particular focus on tools that can utilize gradient in- formation: the derivative of flutter boundaries with respect to random parametric inputs. Results are shown for two test cases: the AGARD 445.6 case and the Common Research Model, and the strength and weaknesses of each UQ tool are compared in terms of cost and accuracy. Two of the UQ tools used here require gradients as part of a search process, but the other two are sample-based, and are able to leverage the existence of gradients for improved performance.

Multi-model Monte Carlo methods, such as multi-level Monte Carlo (MLMC) and multifidelity Monte Carlo (MFMC), allow for efficient estimation of the expectation of a quantity of interest given a set of models of varying fidelities. Recently, it was shown that the MLMC and MFMC estimators are both instances of the approximate control variates (ACV) framework [Gorodetsky et al. 2020]. In that same work, it was also shown that hand-tailored ACV estimators could outperform MLMC and MFMC for a variety of model scenarios. Because there is no reason to believe that these hand-tailored estimators are the best among a myriad of possible ACV estimators, a more general approach to estimator construction is pursued in this work. First, a general form of the ACV estimator variance is formulated. Then, the formulation is utilized to generate parametrically-defined estimators. These parametrically-defined estimators allow for an optimization to be pursued over a larger domain of possible ACV estimators. The parametrically-defined estimators are tested on a large set of model scenarios, and it is found that the broader search domain enabled by parametrically-defined estimators leads to greater variance reduction.

Predicting landing location and other quantities of interest (QoIs) for entry, descent, and landing (EDL) applications requires a viable uncertainty propagation method that accounts for uncertainties in aerodynamics, atmosphere, mass properties, etc. While Monte Carlo (MC) simulation is the de facto standard and produces robust and unbiased statistical estimators, it is often computationally intractable for expensive, high-fidelity models. Low-fidelity models are commonly constructed to replace the high-fidelity model in MC simulation for speedup, but at the expense of accuracy and unbiasedness. Emerging multi-model MC methods are bridging this gap by combining predictions from two or more models of varying fidelity and computational cost for efficient and unbiased uncertainty propagation. This work establishes a proof of concept for using multi-model MC to increase the speed and precision of trajectory simulation for EDL. It is shown that combining a high-fidelity EDL model with low-fidelity models (e.g., data-driven, reduced physics) in this manner has the potential to yield significant efficiency and accuracy gains for certain EDL QoIs versus a standard MC approach. Moreover, the unbiasedness of multi-model MC predictions is highlighted by showing increased accuracy versus an approach that leverages a low-fidelity model alone.

View Video Presentation: https://doi.org/10.2514/6.2022-1099.vid Multi-model Monte Carlo methods have been demonstrated to be an efficient and accurate alternative to standard Monte Carlo (MC) in the model-based propagation of uncertainty in entry, descent, and landing (EDL) applications. These multi-model MC methods fuse predictions from low-fidelity models with the high-fidelity EDL model of interest to produce unbiased statistics with a fraction of the computational cost. The accuracy and efficiency of the multi-model MC methods are dependent upon the magnitude of correlations of the low-fidelity models with the high-fidelity model, but also upon the correlation among the low-fidelity models, and their relative computational cost. Because of this layer of complexity, the question of how to optimally select the set of low-fidelity models has remained open. In this work, methods for optimal model construction and tuning are investigated as a means to increase the speed and precision of trajectory simulation for EDL. Specifically, the focus is on the inclusion of low-fidelity model tuning within the sample allocation optimization that accompanies multi-model MC methods. Preliminary results indicate that low-fidelity model tuning can significantly improve efficiency and precision of trajectory simulations and provide an increased edge to multi-model MC methods when compared to standard MC. The challenges and potential benefits to exploring a fully iterative and comprehensive optimization strategy in future work are highlighted.

An efficient approach for topology optimization under uncertainty is presented. Stochastic reduced order models (SROMs) are leveraged for the modeling and propagation of uncertainties within a robust topology optimization (RTO) formulation. The SROM approach provides an alternative to existing uncertainty quantification methods and yields a substantial improvement in efficiency over a classical Monte Carlo based approach while retaining similar accuracy when representing the uncertainty in system parameters. In particular, random input parameters can be discrete or continuous and specified either analytically (standard distributions) or numerically (dataset samples). Furthermore, multiple random quantities need not be treated as uncorrelated; an SROM can seamlessly model random vectors with arbitrary correlation structure. The nonintrusive nature of the SROM method yields an implementation that can be seen as a drop‐in replacement for a simple RTO approach that leverages Monte Carlo simulation and is therefore straightforward to implement in existing topology optimization software. The proposed approach is demonstrated in the context of structural topology optimization with uncertainty in applied loads. Several numerical results are presented, covering a range of uncertainty distributions that illustrate the flexibility afforded by the general SROM method, while highlighting the efficiency and accuracy achieved in uncertainty propagation.

The Digital Twin (DT) concept has the potential to revolutionize the way systems and their components are designed, managed, maintained, and operated across a vast number of fields from engineering to healthcare. The focus of this work is the implementation of DT for the health management of fatigue critical structures. This paper is the second part of a two-part series. The first of the series demonstrated the use of multi-scale, initiation-to-failure crack growth modeling to form non-deterministic predictions of fatigue life. In this second part, a general method for reducing uncertainty in fatigue life predictions is presented that couples in-situ diagnostics and prognostics in a probabilistic framework. Monte Carlo methods and high-fidelity finite element models are used to (i) generate probabilistic estimates of crack state throughout the life of a geometrically-complex test specimen and (ii) predict fatigue life with decreasing uncertainty as more of these diagnoses are obtained. The ability to predict accurately and in the presence of uncertainty is demonstrated, suggesting that the proposed DT method is feasible for fatigue life prognosis and should be pursued further with a focus on increasing application realism.

Currently, there are several models in the literature, such as kinetic, microstructural, and mass transport, that describe a Li-oxygen battery’s discharge behavior. Many of these models are calibrated and tested at low current densities and are not easily transferable to higher current densities (local current density exceeding 1μA/cm2). Even at low current densities, there is no quantitative method for a researcher to choose a reaction kinetic model such as classical Butler-Volmer and its derivatives, or modified Marcus-Hush-Chidsey, a resistance model for lithium peroxide such as electron transport via tunneling or linear resistivity, a surface coverage model (lithium peroxide growth) such as partial coverage or full coverage, and mass transport model (a review of these models is discussed in Ref. [1]). Also, it is time-consuming to test different models at high current density (≥1C) due to a lack of well-tested models and well-calibrated model parameters. For this presentation, we will develop an analytical model, which acts as a surrogate model for a full finite element model to predict discharge time and discharge voltage. Next, we use an uncertainty quantifying technique called reduced-order stochastic optimization [2, 3] to determine the uncertainty in model parameters for rate kinetics, lithium peroxide resistivity, and parasitic resistance. Finally, a finite element simulation is performed to determine the error introduced by the surrogate model (from various assumptions and simplifications) and its influence on the uncertainty in the model parameters. Our preliminary polarization analysis (overvoltage vs. current density) shows that the resistance models, coupled with Butler-Volmer kinetics, cannot describe the voltage polarization observed in the experiments at high current densities. In order to quantify and rank models describing the lithium peroxide deposition, first, an analytical model is developed to account for discharge product distribution, concentration polarization, and voltage polarization as a function of time, based on the previous work on aqueous electrolytes, presented in Ref. [4]. The analytical model considers reaction kinetics, oxygen mass transport, voltage polarization from lithium peroxide deposition as a conformal coating, and the change in porosity, lithium peroxide thickness, and the active surface is assumed to be uniform throughout the cathode. In order to rank the kinetic and resistance models, the uncertainty in the model parameters is assumed using a probability distribution function, and the resulting statistics are gathered using the Stochastic Reduced Order Model (SROM) approach. SROM is a computationally lite approach that uses significantly less number of samples compared to Monte Carlo to represent the input distribution. Since the number of samples required for Monte Carlo increases with model complexity, which significantly increases the computational time and resources required for full finite element simulations. The SROM approach can reduce the computational requirement by at least ten times without introducing algorithm-based error into the analysis. Although the analytical model incorporates the important aspects to simulate a battery, it neglects the complexity of spatial distribution and temporal variation of the discharge product and the complicated relationship between microstructure and mass transport of oxygen and lithium ions. These neglected interactions introduce a model error in the final parameter estimation and need to be accounted for in order to rank various models. The finite element model (FEM) is used to quantify the error in the surrogate model because of its low fidelity. The FEM model is implemented in COMSOL and considers porous electrode theory for the cathode, concentrated theory for the electrolyte, a resistance model for the discharge products, a kinetic model, Fickian diffusion for the oxygen transport, and an oxygen dissolution model [5]. References: 1. Tan, P., Kong, W., et al. (2017), Prog Energ Combust. (62) 155–189. 2. Wang, H. & Sheen, D. (2015), Prog Energ Combust. (47), 1-31. 3. Sarkar, S., Warner, J., et al. (2014), Corrosion Science (80), 257–268. 4. Mehta, M. & Andrei, P. (2015), J. Power Sources. (286), 299-308. 5. Mehta, M., Knudsen, K. et al. (2019), Meeting Abstracts MA2019-01. (2), 347–347.

Crystal plasticity constitutive models are frequently used with finite elements for modeling metallic grain-scale phenomena. The accuracy of these models is directly a function of the calibrated parameters, which fully define a crystal plasticity model. A number of techniques exist for the calibration of these parameters. In the current study, a comparison of results using deterministic and non-deterministic calibration methods is made. Additionally, the effect of the type of measured data on calibrated material parameters, global (homogenized) or local, is also presented. Included in the study is a new approach to parameter calibration based on combined digital image correlation and high angular resolution electron backscatter diffraction. Utilizing data from these experimental techniques allows for local evaluation of both strain and relative stress: essentially giving stress-strain curves from numerous point locations in a single coupon. The overall result is that calibration based on sub-grain-scale measurements is preferable when sub-grain-scale phenomena are of primary interest.

While standard generative adversarial networks (GANs) rely solely on training data to learn unknown probability distributions, physics-informed GANs (PI-GANs) encode physical laws in the form of stochastic partial differential equations (PDEs) using auto differentiation. By relating observed data to unobserved quantities of interest through PDEs, PI-GANs allow for the estimation of underlying probability distributions without their direct measurement (i.e. inverse problems). The scalable nature of GANs allows high-dimensional, spatially-dependent probability distributions (i.e., random fields) to be inferred, while incorporating prior information through PDEs allows the training datasets to be relatively small. In this work, PI-GANs are demonstrated for the application of elastic modulus estimation in mechanical testing. Given measured deformation data, the underlying probability distribution of spatially-varying elastic modulus (stiffness) is learned. Two feed-forward deep neural network generators are used to model the deformation and material stiffness across a two dimensional domain. Wasserstein GANs with gradient penalty are employed for enhanced stability. In the absence of explicit training data, it is demonstrated that the PI-GAN learns to generate realistic, physically-admissible realizations of material stiffness by incorporating the PDE that relates it to the measured deformation. It is shown that the statistics (mean, standard deviation, point-wise distributions, correlation length) of these generated stiffness samples have good agreement with the true distribution.

Implementation of Finite Elements by NASA (ScIFEN) is a software package developed to solve complex computational materials and structures problems using the finite element method (FEM). In this paper, we describe optimization techniques to speed up the linear solver computation that occurs within the ScIFEN application. We consider GPUs from two different vendors, NVIDIA and AMD as our target platforms for optimization and highlight differences in performance and optimization techniques. The NVIDIA GPU Volta V100 is used in the Summit system deployed at Oak Ridge National Laboratory, and the new exascale system, Frontier, will be using AMD Radeon Instinct GPU. We evaluated the performance of various optimization techniques on test matrices, ranging in size from 100K to 4M, that are representative of ScIFEN applications. The linear solver computation is memory-bound on both GPUs. Our experiments show that on the NVIDIA GPU we obtained up to 79% of the theoretical peak bandwidth, while the AMD GPU achieved 59%. Overall, the NVIDIA V100 GPU outperforms the 1 AMD MI 25 GPU . We observed an overall speedup of up to 37X on an NVIDIA V100 compared to an Intel Skylake 12-core machine. The solver for a 4M degree of freedom system took under 2.5 seconds.