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

TL;DR

This paper introduces a data-driven operator inference approach for creating fast, parameterized reduced-order models of time-dependent PDEs, enabling efficient simulations for uncertainty quantification and inverse problems.

Contribution

It embeds parametric PDE structure into reduced models and learns operators via linear regression, enhancing speed and flexibility over traditional methods.

Findings

Effective for heat equation and neuron model

Handles parametric variations efficiently

Addresses numerical stability and regularization

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.