DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs

TL;DR

DGNet is proposed, a discrete Green network for data-efficient learning of spatiotemporal PDEs that embeds the superposition principle into the hybrid physics-neural architecture, which reduces the burden of learning physical priors from data, thereby improving sample efficiency.

Abstract

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically require large numbers of training trajectories, while high-fidelity PDE data are expensive to generate. Under limited data, their performance degrades substantially, highlighting their low data efficiency. A key reason is that PDE dynamics embody strong structural inductive biases that are not explicitly encoded in neural architectures, forcing models to learn fundamental physical structure from data. A particularly salient manifestation of this inefficiency is poor generalization to unseen source terms. In this work, we revisit Green's function theory-a cornerstone of PDE theory-as a principled source of structural inductive bias for PDE learning. Based on this insight, we propose DGNet, a discrete Green network for data-efficient learning of spatiotemporal PDEs. The key idea is to transform the Green's function into a graph-based discrete formulation, and embed the superposition principle into the hybrid physics-neural architecture, which reduces the burden of learning physical priors from data, thereby improving sample efficiency. Across diverse spatiotemporal PDE scenarios, DGNet consistently achieves state-of-the-art accuracy using only tens of training trajectories. Moreover, it exhibits robust zero-shot generalization to unseen source terms, serving as a stress test that highlights its data-efficient structural design.

Figures (18)

• Figure 1: Overview of DGNet architecture. The model centers on a hybrid operator $\mathbf{L} = \mathbf{L}_{\text{physics}} + \mathbf{L}_{\text{neural}}$, where $\mathbf{L}_{\text{physics}}$ encodes gradient and Laplacian discretizations and $\mathbf{L}_{\text{neural}}$ is a GNN-based correction for mesh-induced errors. This operator is integrated into the discrete Green’s function update (Eq. \ref{['eq:discrete_green']}), which naturally combines system evolution with source-term response and provides structural inductive bias for data-efficient learning.
• Figure 2: Geometric variables for the discrete operator.
• Figure 3: Visualization of prediction results on classical PDE scenarios. Rows from top to bottom correspond to the Allen--Cahn, FitzHugh--Nagumo, and Fisher--KPP equations, respectively.
• Figure 4: Visualization of prediction results on scenarios with complex geometric domains. Rows from top to bottom correspond to the cylinder, sediment, and complex obstacle cases, respectively.
• Figure 5: Visualization of generalization performance on the laser heat scenario with unseen sources.

Theorems & Definitions (2)

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