PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems

TL;DR

PhyMPGN tackles the challenge of predicting spatiotemporal PDE dynamics on irregular meshes with limited training data by integrating a physics-encoded graph neural network with a learnable Laplace block and a boundary-condition padding scheme. Temporal marching is performed via a second-order Runge-Kutta-like integration, while the Laplace block encodes a discrete Laplace-Beltrami operator to constrain the solution space to physically feasible diffusion behavior. The model demonstrates strong performance across Burgers', FitzHugh–Nagumo, Gray-Scott, and cylinder flow, achieving state-of-the-art accuracy, robust generalization to unseen initial conditions and boundary conditions, and favorable computational efficiency compared to both neural and classical solvers. These results indicate significant potential for rapid, accurate PDE simulations on complex domains with limited labeled data, and set the groundwork for extensions to more complex diffusion terms and 3D geometries.

Abstract

Solving partial differential equations (PDEs) serves as a cornerstone for modeling complex dynamical systems. Recent progresses have demonstrated grand benefits of data-driven neural-based models for predicting spatiotemporal dynamics (e.g., tremendous speedup gain compared with classical numerical methods). However, most existing neural models rely on rich training data, have limited extrapolation and generalization abilities, and suffer to produce precise or reliable physical prediction under intricate conditions (e.g., irregular mesh or geometry, complex boundary conditions, diverse PDE parameters, etc.). To this end, we propose a new graph learning approach, namely, Physics-encoded Message Passing Graph Network (PhyMPGN), to model spatiotemporal PDE systems on irregular meshes given small training datasets. Specifically, we incorporate a GNN into a numerical integrator to approximate the temporal marching of spatiotemporal dynamics for a given PDE system. Considering that many physical phenomena are governed by diffusion processes, we further design a learnable Laplace block, which encodes the discrete Laplace-Beltrami operator, to aid and guide the GNN learning in a physically feasible solution space. A boundary condition padding strategy is also designed to improve the model convergence and accuracy. Extensive experiments demonstrate that PhyMPGN is capable of accurately predicting various types of spatiotemporal dynamics on coarse unstructured meshes, consistently achieves the state-of-the-art results, and outperforms other baselines with considerable gains.

Figures (15)

• Figure 1: (a) Model with the second-order Runge-Kutta scheme. (b) NN block consists of two parts: a GNN block followed the Encode-Process-Decode framework and a Laplace block with the correction architecture. Mesh Laplace in Laplace block denotes the discrete Laplace-Beltrami operator in geometric mathematics. The other three modules, $\text{MLP}_\alpha, \text{MLP}_\beta$, and MPNNs, constitutes the lightweight learnable network for correction. (c) The MPNNs module in GNN block consists of $L$ identical MPNN layers. Residual connection and padding in latent space are applied in the first $L-1$ layers, excluding the last layer.
• Figure 2: (a) Four discrete nodes ($i$, $j$, $p$, and $q$) and five edges connecting them. The two angles opposite to the edge $(i, j)$ are $\alpha_{ij}$ and $\beta_{ij}$. (b) The blue region denotes the area of the polygon formed by connecting the circumcenters (e.g., $c_1, \ c_2$, etc.) of the triangles around node $i$.
• Figure 3: Diagram of boundary condition (BC) padding.
• Figure 4: The error distribution (left) over time steps, correlation curves (medium), and predicted snapshots (right) at the last time step. (a) Burgers' equation. (b) FN equation. (c) GS equation. All systems of these PDEs have periodic BCs.
• Figure 5: Four scenarios with different domains and hybrid BCs.