Elizabeth Qian

Jakob Scheffels, Elizabeth Qian, Iason Papaioannou et al.

Bayesian inverse problems use data to update a prior probability distribution on uncertain parameter values to a posterior distribution. Such problems arise in many structural engineering applications, but computational solution of Bayesian inverse problems is often expensive because standard solution approaches require many evaluations of the forward model mapping the parameter value to predicted observations. In many settings, this forward model is expensive because it requires the solution of a high-dimensional discretization of a partial differential equation. However, Bayesian inverse problems often exhibit low-dimensional structure because the available data are primarily informative (relative to the prior) in a low-dimensional subspace, sometimes called the likelihood-informed subspace (LIS). This paper proposes a new projection-based model reduction method for static linear systems that exploits this low-dimensional structure in the setting where the unknown parameter is the right-hand-side forcing, giving rise to a linear inverse problem. The proposed method projects the governing partial differential equation onto the likelihood-informed subspace, yielding a computationally efficient reduced model that can be used to accelerate the solution of the inverse problem and subsequent downstream computations. Numerical experiments on two structural engineering model problems demonstrate that the proposed approach can successfully exploit the intrinsic low-dimensionality of the problem, obtaining relative errors in O(10^{-10}) in the inverse problem solution with a 10x and 100x lower-dimensional model, respectively.

Tomoki Koike, Prakash Mohan, Marc T. Henry de Frahan et al.

High-performance computing enables simulation of high-dimensional physical systems, but downstream analyses such as inverse problems and control remain computationally expensive, motivating model order reduction (MOR) to construct efficient low-dimensional surrogates. Proper Orthogonal Decomposition (POD), a widely adopted data-driven MOR method, projects dynamics onto linear subspaces spanned by the most energetic modes. However, POD struggles for problems with slowly decaying Kolmogorov \(n\)-widths, such as advection-dominated and turbulent flows, requiring many modes for accurate reconstruction. Moreover, energy-based selection can discard crucial low-energy modes needed to capture small-scale features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy mode selection achieve better reconstruction with fewer modes. However, these methods fix the nonlinear mapping form a priori, limiting expressivity. Conversely, neural network (NN) manifolds offer greater expressivity but employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding via POD modes and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through residual connections with linear skip layers, to simultaneously select informative POD modes and learn a nonlinear mapping that minimizes reconstruction error. On benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number \(Re_τ=5200\), we reduce reconstruction error by 51--78\% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

Qiqi Wang, Steven Gomez, Patrick Blonigan et al.

We present a reformulation of unsteady turbulent flow simulations. The initial condition is relaxed and information is allowed to propagate both forward and backward in time. Simulations of chaotic dynamical systems with this reformulation can be proven to be well-conditioned time domain boundary value problems. The reformulation can enable scalable parallel-in-time simulation of turbulent flows.

Josie König, Elizabeth Qian, Melina A. Freitag

Bayesian inverse problems use observed data to update a prior probability distribution for an unknown state or parameter of a scientific system to a posterior distribution conditioned on the data. In many applications, the unknown parameter is high-dimensional, making computation of the posterior expensive due to the need to sample in a high-dimensional space and the need to evaluate an expensive high-dimensional forward model relating the unknown parameter to the data. However, inverse problems often exhibit low-dimensional structure due to the fact that the available data are only informative in a low-dimensional subspace of the parameter space. Dimension reduction approaches exploit this structure by restricting inference to the low-dimensional subspace informed by the data, which can be sampled more efficiently. Further computational cost reductions can be achieved by replacing expensive high-dimensional forward models with cheaper lower-dimensional reduced models. In this work, we propose new dimension and model reduction approaches for linear Bayesian inverse problems with rank-deficient prior covariances, which arise in many practical inference settings. The dimension reduction approach is applicable to general linear Bayesian inverse problems whereas the model reduction approaches are specific to the problem of inferring the initial condition of a linear dynamical system. We provide theoretical approximation guarantees as well as numerical experiments demonstrating the accuracy and efficiency of the proposed approaches.

Atticus Rex, Elizabeth Qian, David Peterson

Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data. Supervised machine learning methods have demonstrated promise in learning efficient surrogate models that can (partially) replace expensive high-fidelity models, making many-query analyses, such as optimization, uncertainty quantification, and inference, tractable. However, when training data must be obtained through the evaluation of an expensive model or experiment, the amount of training data that can be obtained is often limited, which can make learned surrogate models unreliable. However, in many engineering and scientific settings, cheaper \emph{low-fidelity} models may be available, for example arising from simplified physics modeling or coarse grids. These models may be used to generate additional low-fidelity training data. The goal of \emph{multifidelity} machine learning is to use both high- and low-fidelity training data to learn a surrogate model which is cheaper to evaluate than the high-fidelity model, but more accurate than any available low-fidelity model. This work proposes a new multifidelity training approach for Gaussian process regression which uses low-fidelity data to define additional features that augment the input space of the learned model. The approach unites desirable properties from two separate classes of existing multifidelity GPR approaches, cokriging and autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

Tomoki Koike, Elizabeth Qian

In the design and operation of complex dynamical systems, it is essential to ensure that all state trajectories of the dynamical system converge to a desired equilibrium within a guaranteed stability region. Yet, for many practical systems -- especially in aerospace -- this region cannot be determined a priori and is often challenging to compute. One of the most common methods for computing the stability region is to identify a Lyapunov function. A Lyapunov function is a positive function whose time derivative along system trajectories is non-positive, which provides a sufficient condition for stability and characterizes an estimated stability region. However, existing methods of characterizing a stability region via a Lyapunov function often rely on explicit knowledge of the system governing equations. In this work, we present a new physics-informed machine learning method of characterizing an estimated stability region by inferring a Lyapunov function from system trajectory data that treats the dynamical system as a black box and does not require explicit knowledge of the system governing equations. In our presented Lyapunov function Inference method (LyapInf), we propose a quadratic form for the unknown Lyapunov function and fit the unknown quadratic operator to system trajectory data by minimizing the average residual of the Zubov equation, a first-order partial differential equation whose solution yields a Lyapunov function. The inferred quadratic Lyapunov function can then characterize an ellipsoidal estimate of the stability region. Numerical results on benchmark examples demonstrate that our physics-informed stability analysis method successfully characterizes a near-maximal ellipsoid of the system stability region associated with the inferred Lyapunov function without requiring knowledge of the system governing equations.

Tomoki Koike, Elizabeth Qian

Many-query computations, in which a computational model for an engineering system must be evaluated many times, are crucial in design and control. For systems governed by partial differential equations (PDEs), typical high-fidelity numerical models are high-dimensional and too computationally expensive for the many-query setting. Thus, efficient surrogate models are required to enable low-cost computations in design and control. This work presents a physics-preserving reduced model learning approach that targets PDEs whose quadratic operators preserve energy, such as those arising in governing equations in many fluids problems. The approach is based on the Operator Inference method, which fits reduced model operators to state snapshot and time derivative data in a least-squares sense. However, Operator Inference does not generally learn a reduced quadratic operator with the energy-preserving property of the original PDE. Thus, we propose a new energy-preserving Operator Inference (EP-OpInf) approach, which imposes this structure on the learned reduced model via constrained optimization. Numerical results using the viscous Burgers' and Kuramoto-Sivashinksy equation (KSE) demonstrate that EP-OpInf learns efficient and accurate reduced models that retain this energy-preserving structure.

Elizabeth Qian, Dayoung Kang, Vignesh Sella et al.

Machine learning (ML) methods, which fit to data the parameters of a given parameterized model class, have garnered significant interest as potential methods for learning surrogate models for complex engineering systems for which traditional simulation is expensive. However, in many scientific and engineering settings, generating high-fidelity data on which to train ML models is expensive, and the available budget for generating training data is limited, so that high-fidelity training data are scarce. ML models trained on scarce data have high variance, resulting in poor expected generalization performance. We propose a new multifidelity training approach for scientific machine learning via linear regression that exploits the scientific context where data of varying fidelities and costs are available: for example, high-fidelity data may be generated by an expensive fully resolved physics simulation whereas lower-fidelity data may arise from a cheaper model based on simplifying assumptions. We use the multifidelity data within an approximate control variate framework to define new multifidelity Monte Carlo estimators for linear regression models. We provide bias and variance analysis of our new estimators that guarantee the approach's accuracy and improved robustness to scarce high-fidelity data. Numerical results demonstrate that our multifidelity training approach achieves similar accuracy to the standard high-fidelity only approach with orders-of-magnitude reduced high-fidelity data requirements.

Elizabeth Qian, Ionut-Gabriel Farcas, Karen Willcox

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an enabling technology for many computational algorithms used in engineering settings. Our formulation generalizes to the function space PDE setting the Operator Inference method previously developed in [B. Peherstorfer and K. Willcox, Data-driven operator inference for non-intrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering, 306 (2016)] for systems governed by ordinary differential equations. The method brings together two main elements. First, ideas from projection-based model reduction are used to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. Second, supervised machine learning tools are used to infer from data the reduced operators of this physics-informed parametrization. For systems whose governing PDEs contain more general (non-polynomial) nonlinearities, the learned model performance can be improved through the use of lifting variable transformations, which expose polynomial structure in the PDE. The proposed method is demonstrated on two examples: a heat equation model problem that demonstrates the benefits of the function space formulation in terms of consistency with the underlying continuous truth, and a three-dimensional combustion simulation with over 18 million degrees of freedom, for which the learned reduced models achieve accurate predictions with a dimension reduction of five orders of magnitude and model runtime reduction of up to nine orders of magnitude.

Elizabeth Qian, Boris Kramer, Benjamin Peherstorfer et al.

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system's governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh-Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.