Data-driven identification of parametric partial differential equations

TL;DR

This paper introduces a data-driven approach for discovering parametric PDEs using group sparsity and ridge regression, enabling the identification of equations with complex parametric dependencies from observed data.

Contribution

It extends existing PDE identification methods to include parametric dependencies and demonstrates the superiority of group ridge regression over group LASSO.

Findings

Group sequentially thresholded ridge regression outperforms group LASSO.

Method successfully identifies PDE terms and parameters in noisy data.

Applicable to canonical models with complex dependencies.

Abstract

In this work we present a data-driven method for the discovery of parametric partial differential equations (PDEs), thus allowing one to disambiguate between the underlying evolution equations and their parametric dependencies. Group sparsity is used to ensure parsimonious representations of observed dynamics in the form of a parametric PDE, while also allowing the coefficients to have arbitrary time series, or spatial dependence. This work builds on previous methods for the identification of constant coefficient PDEs, expanding the field to include a new class of equations which until now have eluded machine learning based identification methods. We show that group sequentially thresholded ridge regression outperforms group LASSO in identifying the fewest terms in the PDE along with their parametric dependency. The method is demonstrated on four canonical models with and without the introduction of noise.