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Elizabeth Qian

Publications and source records attributed to Elizabeth Qian.

15 recordsLinked to original sources
arXiv2026

Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span. Such linear methods struggle, however, for data with slowly decaying Kolmogorov $n$-widths, such as advection-dominated and turbulent flows, which require many modes for accurate reconstruction; moreover, energy-based truncation can discard 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, but fix the form of the nonlinear mapping a priori, limiting expressivity. In contrast, neural network (NN) manifolds offer greater expressivity yet employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through a residual connection with a linear skip layer, to simultaneously select informative 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 $Re_\tau = 5200$, our method reduces reconstruction error by 51-78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

physics.flu-dyn↗
arXiv2026

Multifidelity-Augmented Gaussian Process Inputs for Surrogate Modeling from Scarce Data

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. In many engineering and scientific settings, cheaper 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 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. Similarly to cokriging estimators, the proposed approach conditions the high-fidelity surrogate model on the predictions of all available low-fidelity surrogate models, while benefiting from the computational efficiency of autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

stat.ML↗
arXiv2026

Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems

Projection-based model reduction enables efficient simulation of complex dynamical systems by constructing low-dimensional surrogate models from high-dimensional data. The Operator Inference (OpInf) approach learns such reduced surrogate models through a two-step process: constructing a low-dimensional basis via Singular Value Decomposition (SVD) to compress the data, then solving a linear least-squares (LS) problem to infer reduced operators that govern the dynamics in this compressed space, all without access to the underlying code or full model operators, i.e., non-intrusively. Traditional OpInf operates as a batch learning method, where both the SVD and LS steps process all data simultaneously. This poses a barrier to deployment of the approach on large-scale applications where dataset sizes prevent the loading of all data into memory at once. Additionally, the traditional batch approach does not naturally allow model updates using new data acquired during online computation. To address these limitations, we propose Streaming OpInf, which learns reduced models from sequentially arriving data streams. Our approach employs incremental SVD for adaptive basis construction and recursive LS for streaming operator updates, eliminating the need to store complete data sets while enabling online model adaptation. The approach can flexibly combine different choices of streaming algorithms for numerical linear algebra: we systematically explore the impact of these choices both analytically and numerically to identify effective combinations for accurate reduced model learning. Numerical experiments on benchmark problems and a large-scale turbulent channel flow demonstrate that Streaming OpInf achieves accuracy comparable to batch OpInf while reducing memory requirements by over 99% and enabling dimension reductions exceeding 31,000x, resulting in orders-of-magnitude faster predictions.

math.NA↗
arXiv2025

Physics-Informed Machine Learning for Characterizing System Stability

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.

cs.LG↗
arXiv2025

Likelihood-informed Model Reduction for Bayesian Inference of Static Structural Loads

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.

math.NA↗
arXiv2025

An ensemble Kalman approach to randomized maximum likelihood estimation

This work proposes ensemble Kalman randomized maximum likelihood estimation, a new derivative-free method for performing randomized maximum likelihood estimation, which is a method that can be used to generate approximate samples from posterior distributions in Bayesian inverse problems. The new method has connections to ensemble Kalman inversion and works by evolving an ensemble so that each ensemble member solves an instance of a randomly perturbed optimization problem. Linear analysis demonstrates that ensemble members converge exponentially fast to randomized maximum likelihood estimators and, furthermore, that the new method produces samples from the Bayesian posterior when applied to a suitably regularized optimization problem. The method requires that the forward operator, relating the unknown parameter to the data, be evaluated once per iteration per ensemble member, which can be prohibitively expensive when the forward model requires the evolution of a high-dimensional dynamical system. We propose a strategy for making the proposed method tractable in this setting based on a balanced truncation model reduction method tailored to the Bayesian smoothing problem. Theoretical results show near-optimality of this model reduction approach via convergence to an optimal approximation of the posterior covariance as a low-rank update to the prior covariance. Numerical experiments verify theoretical results and illustrate computational acceleration through model reduction.

math.NA↗
arXiv2025

Dimension and model reduction approaches for linear Bayesian inverse problems with rank-deficient prior covariances

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.

math.NA↗
arXiv2024

The Fundamental Subspaces of Ensemble Kalman Inversion

Ensemble Kalman Inversion (EKI) methods are a family of iterative methods for solving weighted least-squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI requires only evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI's convergence properties in terms of ``fundamental subspaces'' analogous to Strang's fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.

math.NA↗
arXiv2024

Multifidelity linear regression for scientific machine learning from scarce data

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.

stat.ML↗
arXiv2024

Energy-Preserving Reduced Operator Inference for Efficient Design and Control

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.

math.NA↗
arXiv2022

The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks

The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential equations (PDEs). Surrogate modeling is an enabling methodology for many-query computations in science and engineering, which include iterative methods in optimization and sampling methods in uncertainty quantification. Over the last few years, several approaches to surrogate modeling for PDEs using neural networks have emerged, motivated by successes in using neural networks to approximate nonlinear maps in other areas. In principle, the relative merits of these different approaches can be evaluated by understanding, for each one, the cost required to achieve a given level of accuracy. However, the absence of a complete theory of approximation error for these approaches makes it difficult to assess this cost-accuracy trade-off. The purpose of the paper is to provide a careful numerical study of this issue, comparing a variety of different neural network architectures for operator approximation across a range of problems arising from PDE models in continuum mechanics.

math.NA↗
arXiv2022

Reduced operator inference for nonlinear partial differential equations

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.

math.NA↗
arXiv2021

Model Reduction of Linear Dynamical Systems via Balancing for Bayesian Inference

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions as expressed through the associated Gramians. We introduce Gramian definitions relevant to the inference setting and propose a balanced truncation approach based on these inference Gramians that yield a reduced dynamical system that can be used to cheaply approximate the posterior mean and covariance. Our definitions exploit natural connections between (i) the reachability Gramian and the prior covariance and (ii) the observability Gramian and the Fisher information. The resulting reduced model then inherits stability properties and error bounds from system theoretic considerations, and in some settings yields an optimal posterior covariance approximation. Numerical demonstrations on two benchmark problems in model reduction show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

eess.SY↗
arXiv2020

Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

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.

math.NA↗
arXiv2012

Towards Scalable Parallel-in-Time Turbulent Flow Simulations

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.

physics.flu-dyn↗