Operator Inference for Parametric PDEs

This repository is the source code for the preprint Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference (PDF) by McQuarrie, Khodabakhshi, and Willcox.

BibTeX

@article{mcquarrie2021opinf,
author = {Shane A. McQuarrie and Parisa Khodabakhshi and Karen E. Willcox},
title = {Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference},
journal = {arXiv preprint arXiv:2110.07753},
year = {2021},
}

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 (shown above).

Repository Contents

Heat Equation

• heat.py: defines classes for solving the one-dimensional parametric heat problem with piecewise constant diffusion.
  • HeatSolver: high-fidelity finite difference solver.
  • HeatROM: operator inference reduced-order model solver.

FitzHugh-Nagumo System

• fhn.py: defines classes for solving the FitzHugh-Nagumo neuron model.
  • FHNSolver: high-fidelity finite difference solver.
  • FHNROMSolver: reduced-order model solver.
  • AffineFHNROM: operator inference reduced-order model.
  • AffineFHNROM_Intrusive: reduced-order model from intrusive projection.
• fhn_rom_search.py: script for operator inference hyperparameter search.

Utilities

• config.py: configuration (naming conventions, plot customizations, etc.).
• utils.py: utilities (logging, timing, data management).

Citation

If you find this repository useful, please consider citing our paper:

McQuarrie, S. A., Khodabakhshi, P and Willcox, K. E., Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference. arXiv preprint 2110.07653, 2021.

@article{mcquarrie2021popinf,
    author = {Shane A. McQuarrie and Parisa Khodabakhshi and Karen E. Willcox},
    title = {Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference},
    journal = {arXiv preprint arXiv:2110.07753},
    year = {2021},
}