用图神经网络+符号回归，从噪声多、采样稀疏的数据中发现物理方程。
Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data
- 通过图消息传递建模空间关系，避免局部微分对噪声敏感
- 在波方程、对流扩散方程等测试中成功恢复真实物理规律
- 适合处理实际观测中常见的低质量数据，可解释性强
基于数据驱动的偏微分方程（PDE）发现为从观测数据中揭示控制物理规律提供了新范式。然而，在实际场景中，测量常受噪声污染且采样稀疏，现有依赖数值微分或积分形式的方法面临严峻挑战。本文提出符号图网络（SGN）框架，用于在噪声大、采样稀疏条件下进行PDE发现。SGN不依赖局部微分近似，而是利用图消息传递建模空间相互作用，提供对高频噪声不敏感的非局部表示。在此基础上，通过符号回归模块提取可解释的数学表达式。在波方程、对流扩散方程及不可压缩纳维-斯托克斯方程等基准系统上评估显示，该方法可在不同噪声水平下恢复有意义的控制关系或解形式，相比基线方法在稀疏和噪声环境中表现出更强鲁棒性。结果表明，将图表示与符号回归结合是应对不完美观测下物理规律发现的有效路径。代码已公开于 https://github.com/CXY0112/SGN。
原文摘要 · Abstract (English)
Data-driven discovery of partial differential equations (PDEs) offers a promising paradigm for uncovering governing physical laws from observational data. However, in practical scenarios, measurements are often contaminated by noise and limited by sparse sampling, which poses significant challenges to existing approaches based on numerical differentiation or integral formulations. In this work, we propose a Symbolic Graph Network (SGN) framework for PDE discovery under noisy and sparse conditions. Instead of relying on local differential approximations, SGN leverages graph message passing to model spatial interactions, providing a non-local representation that is less sensitive to high frequency noise. Based on this representation, the learned latent features are further processed by a symbolic regression module to extract interpretable mathematical expressions. We evaluate the proposed method on several benchmark systems, including the wave equation, convection-diffusion equation, and incompressible Navier-Stokes equations. Experimental results show that SGN can recover meaningful governing relations or solution forms under varying noise levels, and demonstrates improved robustness compared to baseline methods in sparse and noisy settings. These results suggest that combining graph-based representations with symbolic regression provides a viable direction for robust data-driven discovery of physical laws from imperfect observations. The code is available at https://github.com/CXY0112/SGN
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