通过平衡引导策略，从数据中发现含小系数项的多尺度非线性方程。
Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms
- 基于主导平衡原理，按项对方程平衡的相对贡献排序
- 在多个测试系统中成功识别出系数极小但动态关键的项
- 适合研究复杂系统中微弱但重要作用的物理机制
近年来数据驱动的控制方程发现取得显著进展，但现有方法在多尺度系统中仍面临挑战，即动态重要的项可能具有小系数。为此，我们提出平衡引导稀疏识别（BG-SINDy），受主导平衡原理启发，将ℓ₀约束的稀疏回归重构为逐项ℓ₂,₀正则化问题，并采用渐进式剪枝策略求解。项的排序依据其对控制方程平衡的相对贡献，而非绝对系数大小。在此准则下，BG-SINDy交替进行最小二乘回归与次要项剔除，从而在系数很小的情况下仍能保留动态关键项。在小色散系数的Korteweg-de Vries方程、消失超粘性的修正Burgers方程、含多个小系数项的修正Kuramoto-Sivashinsky方程及二维反应-扩散系统上的数值实验验证了该方法的有效性。所提方法为发现包含小系数项的控制方程提供了高效途径。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。