Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms

TL;DR

This paper introduces BG-SINDy, a novel method for discovering multiscale nonlinear PDEs with small coefficients by ranking terms based on their contribution to the governing balance, improving detection of significant but small terms.

Contribution

The paper proposes a balance-guided sparse regression approach that effectively identifies small-coefficient terms in multiscale PDEs, addressing limitations of existing methods.

Findings

Successfully identified small-coefficient terms in various PDEs.

Demonstrated effectiveness on KdV, Burgers, Kuramoto--Sivashinsky, and reaction-diffusion systems.

Outperformed traditional methods in preserving dynamically significant small terms.

Abstract

Data-driven discovery of governing equations has advanced significantly in recent years; however, existing methods often struggle in multiscale systems where dynamically significant terms may have small coefficients. Therefore, we propose Balance-Guided SINDy (BG-SINDy) inspired by the principle of dominant balance, which reformulates $\ell_0$-constrained sparse regression as a term-level $\ell_{2,0}$-regularized problem and solves it using a progressive pruning strategy. Terms are ranked according to their relative contributions to the governing equation balance rather than their absolute coefficient magnitudes. Based on this criterion, BG-SINDy alternates between least-squares regression and elimination of negligible terms, thereby preserving dynamically significant terms even when their coefficients are small. Numerical experiments on the Korteweg--de Vries equation with a small dispersion coefficient, a modified Burgers equation with vanishing hyperviscosity, a modified Kuramoto--Sivashinsky equation with multiple small-coefficient terms, and a two-dimensional reaction--diffusion system demonstrate the validity of BG-SINDy in discovering small-coefficient terms. The proposed method thus provides an efficient approach for discovering governing equations that contain small-coefficient terms.