Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

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

Figures (5)

• Figure 1: The overall framework of SGN. The architecture comprises two synergistic components: (1) the Graph Neural Simulator, which functions as a mesh-free discretization engine. It leverages graph message passing to extract robust non-local spatial operators from sparse and noisy observations, aided by auxiliary stabilization strategies; and (2) the Symbolic Inverse Analyzer, which distills these physics-aligned latent features into explicit, parsimonious governing equations.
• Figure 2: Savitzky-Golay filtering. The dashed line exhibits the severe high-frequency jitter of raw noisy observations. In contrast, the solid curves demonstrate the smoothed physical trends recovered by increasing window sizes. This filtering effectively suppresses stochastic noise while preserving geometric structure, providing a stable, derivative-friendly initialization for the GNN simulator.
• Figure 3: Visual comparison of predicted wave fields ($u(x, y, t)$) at time $t=1.0$ under different noise regimes. Top row: low noise (0.2%); Bottom row: severe noise (5.0%). Columns present Ground Truth, SGN (ours), and PDE-NET 2.0 (baseline). While all methods converge under low noise, SGN maintains structural stability and captures the physical wave front even at 5.0% noise, whereas PDE-NET 2.0 exhibits catastrophic numerical divergence (indicated by 'Inf' on a blank background).
• Figure 4: Visual comparison of predicted solutions for the Convection-Diffusion Equation with 0.5% noise. Spatial distributions are shown at two time steps: $t=0.2$ (top row) and $t=2.0$ (bottom row). Columns display Ground Truth, SGN, and PDE-NET 2.0. The figure illustrates the relative degradation of prediction coherent structures at longer simulation times for both learned models compared to Ground Truth.
• Figure 5: Snapshots of the Navier-Stokes flow dynamics under a severe 10% noise level. The figure illustrates the Ground Truth alongside the SGN predictions. Even when the underlying data is heavily obscured by high-frequency artifacts, the proposed SGN architecture flawlessly maintains the structural stability of the fluid flow, achieving highly consistent physical predictions without relying on dense spatial sampling.