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

arXiv 2604.18414 ↗First posted 20 Apr 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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 ℓ0-constrained sparse regression as a term-level ℓ2,0-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.

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References · 16

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
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  5. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 137
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  11. 2021Robust discovery of partial differential equations in complex situationscited by 21
  12. 2023Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Datacited by 16
4 more
  1. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  2. 2025asKAN: Active Subspace embedded Kolmogorov-Arnold Networkcited by 5
  3. 2024Scalable Sparse Regression for Model Discovery: The Fast Lane to Insightcited by 4
  4. 2025An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulencecited by 1

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