Karen Willcox’s research while affiliated with University of Texas at Austin and other places

Ice sheet simulations suffer from vast parametric uncertainties, such as the basal sliding boundary condition or geothermal heat flux. Quantifying the resulting uncertainties in predictions is of utmost importance to support judicious decision-making, but high-fidelity simulations are too expensive to embed within uncertainty quantification (UQ) computations. UQ methods typically employ Monte Carlo simulation to estimate statistics of interest, which requires hundreds (or more) of ice sheet simulations. Cheaper low-fidelity models are readily available (e.g., approximated physics, coarser meshes), but replacing the high-fidelity model with a lower fidelity surrogate introduces bias, which means that UQ results generated with a low-fidelity model cannot be rigorously trusted. Multifidelity UQ retains the high-fidelity model but expands the estimator to shift computations to low-fidelity models, while still guaranteeing an unbiased estimate. Through this exploitation of multiple models, multifidelity estimators guarantee a target accuracy at reduced computational cost. This paper presents a comprehensive multifidelity UQ framework for ice sheet simulations. We present three multifidelity UQ approaches—Multifidelity Monte Carlo, Multilevel Monte Carlo, and the Best Linear Unbiased Estimator—that enable tractable UQ for continental-scale ice sheet simulations. We demonstrate the techniques on a model of the Greenland ice sheet to estimate the 2015-2050 ice mass loss, verify their estimates through comparison with Monte Carlo simulations, and give a comparative performance analysis. For a target accuracy equivalent to 1 mm sea level rise contribution at 95 % confidence, the multifidelity estimators achieve computational speedups of two orders of magnitude.

Design problems in aerospace engineering often require numerous evaluations of expensive-to-evaluate high-fidelity models, resulting in prohibitive computational costs. One way to address the computational cost is through building surrogates, such as deep neural networks (DNNs). However, DNNs may only be an effective surrogate when sufficient evaluations of the high-fidelity model are required such that the up-front training cost is amortized, or in situations that require real-time responses (such as interactive visualizations). Typically, the data requirements for adequately accurate training of DNNs are often impractical for engineering applications. To alleviate this issue, the proposed work utilizes output dimensionality reduction along with information from multiple models of varying fidelities and cost to develop accurate projection-enabled multifidelity neural networks (MF-NNs) with limited training samples. The dimensionality reduction leads to a more parsimonious network and the multifidelity aspect adds more training data from lower-cost, lower-fidelity models. Three approaches for MF-NNs that leverage proper orthogonal decomposition based projections are introduced: (i) pre-training method, (ii) additive method, and (iii) multi-step method. The MF-NN is applied to approximate the optimal design of 2D aerodynamic airfoils given the performance and design requirements. The MF-NN leads to ∼ 27% computational cost reduction compared to single-fidelity neural networks at the same accuracy (90%), with the multi-step approach performing the best for this application.

Ice sheet simulations suffer from vast parametric uncertainties, such as the basal sliding boundary condition or geothermal heat flux. Quantifying the resulting uncertainties in predictions is of utmost importance to support judicious decision-making, but high-fidelity simulations are too expensive to embed within uncertainty quantification (UQ) computations. UQ methods typically employ Monte Carlo simulation to estimate statistics of interest, which requires hundreds (or more) of ice sheet simulations. Cheaper low-fidelity models are readily available (e.g., approximated physics, coarser meshes), but replacing the high-fidelity model with a lower fidelity surrogate introduces bias, which means that UQ results generated with a low-fidelity model cannot be rigorously trusted. Multifidelity UQ retains the high-fidelity model but expands the estimator to shift computations to low-fidelity models, while still guaranteeing an unbiased estimate. Through this exploitation of multiple models, multifidelity estimators guarantee a target accuracy at reduced computational cost. This paper presents a comprehensive multifidelity UQ framework for ice sheet simulations. We present three multifidelity UQ approaches -- Multifidelity Monte Carlo, Multilevel Monte Carlo, and the Best Linear Unbiased Estimator -- that enable tractable UQ for continental-scale ice sheet simulations. We demonstrate the techniques on a model of the Greenland ice sheet to estimate the 2015-2050 ice mass loss, verify their estimates through comparison with Monte Carlo simulations, and give a comparative performance analysis. For a target accuracy equivalent to 1 mm sea level rise contribution at 95% confidence, the multifidelity estimators achieve computational speedups of two orders of magnitude.

This work develops an efficient real-time inverse formulation for inferring the aerodynamic surface pressures on a hypersonic vehicle from sparse measurements of the structural strain. The approach aims to provide real-time estimates of the aerodynamic loads acting on the vehicle for ground and flight testing, as well as guidance, navigation, and control applications. Specifically, the approach targets hypersonic flight conditions where direct measurement of the surface pressures is challenging due to the harsh aerothermal environment. For problems employing a linear elastic structural model, we show that the inference problem can be posed as a least-squares problem with a linear constraint arising from a finite element discretization of the governing elasticity partial differential equation. Due to the linearity of the problem, an explicit solution is given by the normal equations. Pre-computation of the resulting inverse map enables rapid evaluation of the surface pressure and corresponding integrated quantities, such as the force and moment coefficients. The inverse approach additionally allows for uncertainty quantification, providing insights for theoretical recoverability and robustness to sensor noise. Numerical studies demonstrate the estimator performance for reconstructing the surface pressure field, as well as the force and moment coefficients, for the Initial Concept 3.X (IC3X) conceptual hypersonic vehicle.

Digital twins bring value to mechanical and aerospace systems by speeding up development, reducing risk, predicting issues and reducing sustainment costs. Realizing these benefits at scale requires a structured and intentional approach to digital twin conception, design, development, operation and sustainment. To bring maximal value, a digital twin does not need to be an exquisite virtual replica but instead must be envisioned to be fit for purpose, where the determination of fitness depends on the capability needs and the cost-benefit trade-offs.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Efficient methods for achieving active learning in complex physical systems are essential for achieving the two-way interaction between data and models that underlies DDDAS. This work presents a two-stage multifidelity active learning method for Gaussian-process-based reliability analysis. In the first stage, the method allows for the flexibility of using any single-fidelity acquisition function for failure boundary identification when selecting the next sample location. We demonstrate the generalized multifidelity method using the existing acquisition functions of expected feasibility, U-learning, targeted integrated mean square error acquisition functions, or their a priori Monte Carlo sampled variants. The second stage uses a weighted information-gain-based criterion for the fidelity model selection. The multifidelity method leads to significant computational savings over the single-fidelity versions for real-time reliability analysis involving expensive physical system simulations.