Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

2026-03-23

AI Summary

Accurately discovering governing partial differential equations (PDEs) from noisy and sparse observations remains challenging, as conventional approaches suffer from poor robustness due to their reliance on numerical differentiation. This work proposes a Symbolic Graph Network (SGN) framework that uniquely integrates graph neural networks with symbolic regression. By leveraging graph message passing to model non-local spatial interactions, SGN effectively suppresses noise and extracts interpretable mathematical expressions directly from learned latent features—bypassing the need for local differential approximations. The method demonstrates superior performance over existing techniques on benchmark PDE systems, including the wave equation, convection–diffusion equation, and Navier–Stokes equations, reliably recovering the correct governing equations even under high noise levels or extreme sparsity in the data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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

Problem

• PDE discovery
• noisy data
• sparse data
• robustness
• data-driven discovery

Innovation

• Symbolic Graph Network
• PDE discovery
• graph message passing
• symbolic regression
• noisy sparse data