DGNet: Learning Spatiotemporal PDEs with Discrete Green Networks

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications, yet classical solvers are computationally costly on large or irregular domains. Neural PDE solvers have emerged as an efficient alternative, but they struggle to generalize to unseen source terms, which represent external forcing such as heat generation, body forces, or reaction rates. Since existing models typically mix the source with the system state, they lack a principled mechanism to capture source responses. We propose DGNet, a discrete Green network that explicitly decouples system evolution from source response. The key idea is to transform the classical Green’s function---a cornerstone of PDE theory---into a graph-based discrete formulation, preserving the superposition principle in a computable update rule. To ensure fidelity on irregular meshes, we construct a hybrid operator by combining physics-based discretizations with GNN-based corrections, while a lightweight residual GNN captures dynamics beyond the operator. Across three categories of spatiotemporal PDE scenarios, DGNet consistently achieves state-of-the-art accuracy. In particular, on the most challenging setting with entirely novel source terms, DGNet maintains stable performance while existing approaches collapse.