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

This work presents a non-intrusive reduced-order modeling framework for dynamical systems with spatially localized features characterized by slow singular value decay. The proposed approach builds upon two existing methodologies for reduced and full-order non-intrusive modeling, namely Operator Inference (OpInf) and sparse Full-Order Model (sFOM) inference. We decompose the domain into two complementary subdomains which exhibit fast and slow singular value decay. The dynamics of the subdomain exhibiting slow singular value decay are learned with sFOM while the dynamics with intrinsically low dimensionality on the complementary subdomain are learned with OpInf. The resulting, coupled OpInf-sFOM formulation leverages the computational efficiency of OpInf and the high resolution of sFOM, and thus enables fast non-intrusive predictions for conditions beyond those sampled in the training data set. A novel regularization technique with a closed-form solution based on the Gershgorin disk theorem is introduced to promote stable sFOM and OpInf models. We also provide a data-driven indicator for the subdomain selection and ensure solution smoothness over the interface via a post-processing interpolation step. We evaluate the efficiency of the approach in terms of offline and online speedup through a quantitative, parametric computational cost analysis. We demonstrate the coupled OpInf-sFOM formulation for two test cases: a one-dimensional Burgers' model for which accurate predictions beyond the span of the training snapshots are presented, and a two-dimensional parametric model for the Pine Island Glacier ice thickness dynamics, for which the OpInf-sFOM model achieves an average prediction error on the order of $1 \%$ with an online speedup factor of approximately $8\times$ compared to the numerical simulation.

Verifying the correctness of a digital twin provides a formal guarantee that the digital twin operates as intended. Digital twin verification is challenging due to the presence of uncertainties in the virtual representation, the physical environment, and the bidirectional flow of information between physical and virtual. A further challenge is that a digital twin of a complex system is composed of distributed components. This paper presents a methodology to specify and verify digital twin behavior, translating uncertain processes into a formally verifiable finite state machine. We use the Temporal Logic of Actions (TLA) to create a specification, an implementation abstraction that defines the properties required for correct system behavior. Our approach includes a novel weakening of formal security properties, allowing controlled information leakage while preserving theoretical guarantees. We demonstrate this approach on a digital twin of an unmanned aerial vehicle, verifying synchronization of physical-to-virtual and virtual-to-digital data flows to detect unintended misalignments.

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, 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. Precomputation 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.

This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training datasets. In these settings, standard, single-domain approaches may be too inaccurate or may overfit and hence generalize poorly. Moreover, processing large-scale datasets typically requires significant memory and computing resources, which can render single-domain approaches computationally prohibitive. To address these challenges, we introduce a domain-decomposition formulation into the construction of a data-driven reduced model. In doing so, the basis functions used in the reduced model approximation become localized in space, which can increase the accuracy of the domain-decomposed approximation of the complex dynamics. The decomposition furthermore reduces the memory and computing requirements to process the underlying large-scale training dataset. We demonstrate the effectiveness and scalability of our approach in a large-scale three-dimensional unsteady rotating-detonation rocket engine simulation scenario with more than 75 million degrees of freedom and a sparse training dataset. Our results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure by up to 13% and up to 5% for other key quantities, such as temperature, and fuel, and oxidizer mass fractions. Lastly, our approach decreases the memory requirements for processing by almost a factor of four, which in turn reduces the computing requirements as well.

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. We demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion-reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

This work proposes a mathematical framework to increase the robustness to rare events of digital twins modelled with graphical models. We incorporate probabilistic model-checking and linear programming into a dynamic Bayesian network to enable the construction of risk-averse digital twins. By modeling with a random variable the probability of the asset to transition from one state to another, we define a parametric Markov decision process. By solving this Markov decision process, we compute a policy that defines state-dependent optimal actions to take. To account for rare events connected to failures we leverage risk measures associated with the distribution of the random variables describing the transition probabilities. We refine the optimal policy at every time step resulting in a better trade off between operational costs and performances. We showcase the capabilities of the proposed framework with a structural digital twin of an unmanned aerial vehicle and its adaptive mission replanning.

High-performance computing (HPC) has revolutionized our ability to perform detailed simulations of complex real-world processes. A prominent contemporary example is from aerospace propulsion, where HPC is used for rotating detonation rocket engine (RDRE) simulations in support of the design of next-generation rocket engines; however, these simulations take millions of core hours even on powerful supercomputers, which makes them impractical for engineering tasks like design exploration and risk assessment. Reduced-order models (ROMs) address this limitation by constructing computationally cheap yet sufficiently accurate approximations that serve as surrogates for the high-fidelity model. This paper contributes a new distributed algorithm that achieves fast and scalable construction of predictive physics-based ROMs trained from sparse datasets of extremely large state dimension. The algorithm learns structured physics-based ROMs that approximate the dynamical systems underlying those datasets. This enables model reduction for problems at a scale and complexity that exceeds the capabilities of existing approaches. We demonstrate our algorithm's scalability using up to 2,048 cores on the Frontera supercomputer at the Texas Advanced Computing Center. We focus on a real-world three-dimensional RDRE for which one millisecond of simulated physical time requires one million core hours on a supercomputer. Using a training dataset of 2,536 snapshots each of state dimension 76 million, our distributed algorithm enables the construction of a predictive data-driven reduced model in just 13 seconds on 2,048 cores on Frontera.

The digital twin concept represents an appealing opportunity to advance condition-based and predictive maintenance paradigms for civil engineering systems, thus allowing reduced lifecycle costs, increased system safety, and increased system availability. This work proposes a predictive digital twin approach to the health monitoring, maintenance, and management planning of civil engineering structures. The asset-twin coupled dynamical system is encoded employing a probabilistic graphical model, which allows all relevant sources of uncertainty to be taken into account. In particular, the time-repeating observations-to-decisions flow is modeled using a dynamic Bayesian network. Real-time structural health diagnostics are provided by assimilating sensed data with deep learning models. The digital twin state is continually updated in a sequential Bayesian inference fashion. This is then exploited to inform the optimal planning of maintenance and management actions within a dynamic decision-making framework. A preliminary offline phase involves the population of training datasets through a reduced-order numerical model and the computation of a health-dependent control policy. The strategy is assessed on two synthetic case studies, involving a cantilever beam and a railway bridge, demonstrating the dynamic decision-making capabilities of health-aware digital twins.