WG-IDENT: Weak Group Identification of PDEs with Varying Coefficients

TL;DR

This work tackles data-driven identification of PDEs with spatially varying coefficients from noisy spatiotemporal data. It introduces WG-IDENT, a weak-form, group-sparsity framework that represents spatially varying coefficients via B-spline bases and uses optimized test functions to reduce noise impact. The method combines Successively Denoised Differentiation (SDD) for stable time derivatives, Group Subspace Pursuit (GPSP) for coefficient selection, and a trimming plus Reduction in Residual (RR) pipeline to robustly select the final model. Extensive numerical experiments show superior robustness to noise and reduced hyperparameter sensitivity compared with state-of-the-art approaches, highlighting practical applicability to complex, noisy systems.

Abstract

Partial Differential Equations (PDEs) identification is a data-driven method for mathematical modeling, and has received a lot of attentions recently. The stability and precision in identifying PDE from heavily noisy spatiotemporal data present significant difficulties. This problem becomes even more complex when the coefficients of the PDEs are subject to spatial variation. In this paper, we propose a Weak formulation of Group-sparsity-based framework for IDENTifying PDEs with varying coefficients, called WG-IDENT, to tackle this challenge. Our approach utilizes the weak formulation of PDEs to reduce the impact of noise. We represent test functions and unknown PDE coefficients using B-splines, where the knot vectors of test functions are optimally selected based on spectral analysis of the noisy data. To facilitate feature selection, we propose to integrate group sparse regression with a newly designed group feature trimming technique, called GF-trim, to eliminate unimportant features. Extensive and comparative ablation studies are conducted to validate our proposed method. The proposed method not only demonstrates greater robustness to high noise levels compared to state-of-the-art algorithms but also achieves superior performance while exhibiting reduced sensitivity to hyperparameter selection.

Figures (6)

• Figure 1: The 10 B-spline basis with 512 points among $[-30,30]$ with the periodic boundary condition.
• Figure 2: Resulting test function $\phi_x$ along with reference Gaussian $\rho_\sigma$ with $\sigma = \frac{\sqrt{k}}{2\sqrt{3}}h= \frac{\alpha}{\sqrt{3k}}$.Larger texts
• Figure 3: Burger's equation with diffusion under 0% and 1% noise level with 16 dictionary size by using SGTR method.
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