Adaptation of uncertainty-penalized Bayesian information criterion for parametric partial differential equation discovery

TL;DR

This paper extends the UBIC method to efficiently discover parametric PDEs from noisy data by incorporating PDE uncertainty and spectral analysis, accurately identifying true terms and coefficients without costly simulations.

Contribution

An extended UBIC method is proposed for parametric PDE discovery that leverages PDE uncertainty and spectral analysis, improving accuracy and efficiency in noisy conditions.

Findings

Successfully identifies true PDE terms and coefficients in noisy data.

Accurately captures parametric dependencies with confidence intervals.

Requires no computationally expensive PDE simulations.

Abstract

Data-driven discovery of partial differential equations (PDEs) has emerged as a promising approach for deriving governing physics when domain knowledge about observed data is limited. Despite recent progress, the identification of governing equations and their parametric dependencies using conventional information criteria remains challenging in noisy situations, as the criteria tend to select overly complex PDEs. In this paper, we introduce an extension of the uncertainty-penalized Bayesian information criterion (UBIC), which is adapted to solve parametric PDE discovery problems efficiently without requiring computationally expensive PDE simulations. This extended UBIC uses quantified PDE uncertainty over different temporal or spatial points to prevent overfitting in model selection. The UBIC is computed with data transformation based on power spectral densities to discover the governing parametric PDE that truly captures qualitative features in frequency space with a few significant terms and their parametric dependencies (i.e., the varying PDE coefficients), evaluated with confidence intervals. Numerical experiments on canonical PDEs demonstrate that our extended UBIC can identify the true number of terms and their varying coefficients accurately, even in the presence of noise. The code is available at \url{https://github.com/Pongpisit-Thanasutives/parametric-discovery}.