[论文解读] Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

Xingyu Chen, Junxiu An+2 more|arXiv (Cornell University)|Mar 23, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

SGN 将图神经网络与符号回归相结合，在嘈杂、稀疏采样数据中鲁棒地发现偏微分方程（PDE），提供非局部积分式表示和可解释的闭式表达。它在高噪声和数据稀疏条件下优于强基线。

ABSTRACT

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

研究动机与目标

• Motivate robust PDE discovery when data are noisy and sparsely sampled.
• Propose a mesh-free, non-local graph-based representation of spatial operators for PDE learning.
• Integrate symbolic regression to extract interpretable governing equations from learned features.
• Demonstrate robustness and interpretability on wave, convection–diffusion, and Navier–Stokes systems.

提出的方法

• Represent spatial operators via graph message passing on a spatial graph to form a non-local integral operator.
• Decompose the learned operator into a pairwise message function and an update function to reduce the symbolic regression search space.
• Stabilize training with Savitzky–Golay based warm-starts and adaptive noise injection.
• Use PySR symbolic regression to discover closed-form expressions for the message and update laws, composing them into the final PDE.

实验结果

研究问题

• RQ1 Can a graph-based, mesh-free framework learn robust, non-local spatial operators from noisy, sparse data?
• RQ2 Can symbolic regression recover interpretable governing equations from the latent SGN representations?
• RQ3 How does SGN perform on representative PDEs (wave, convection–diffusion, Navier–Stokes) under varying noise levels and data sparsity?

主要发现

• SGN consistently achieves robust qualitative and quantitative recovery of governing relations under increasing noise and data sparsity.
• On the wave equation, SGN discovers the exact analytical solution form u = sin(πx) sin(πy) cos(t) with high accuracy even at 0.2% noise, while PDE-Net 2.0 tends to fail and produce NaNs at higher noise.
• SGN shows favorable stability compared with PDE-Net 2.0 in coarse grids where local derivative-based methods fail due to noise amplification and truncation errors.
• The framework can recover meaningful physical relations and solutions for convection–diffusion and Navier–Stokes systems, with symbolic expressions that reflect the underlying dynamics rather than purely numerical approximations.
• Auxiliary stabilization (S-G warm-start and noise-adaptive training) helps SGN maintain robust performance across tasks.