Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference

2021/10/14 by Shane A. McQuarrie, McQuarrie, Shane A, Parisa Khodabakhshi +3 · 11 citations
Computer Science · Materials Science · Physics and Astronomy · #35B30 #35R30 #65F22 #Computational Engineering #FOS: Computer and information sciences #FOS: Mathematics #Finance #Magnetic Properties and Applications #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA) #and Science (cs.CE)

paper · doi:10.48550/arxiv.2110.07653

openalex publication_date 2021/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.

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Related

• Reduced operator inference for nonlinear partial differential equations
• Reduced-order modeling for parameterized PDEs via implicit neural representations
• Tensor parametric Hamiltonian operator inference
• Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators
• Robust Data-Driven Discovery of Partial Differential Equations under Uncertainties