用格林函数设计神经网络,仅用几十条数据就高效求解时空偏微分方程。
DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs
- 将格林函数转为图结构离散化,嵌入叠加原理提升物理先验学习效率。
- 仅需数十条训练轨迹即达当前最优精度,且在未见源项下零样本泛化良好。
- 适合低数据量、强物理结构依赖的科学计算场景,如气候模拟与流体建模。
时空偏微分方程(PDE)广泛应用于科学与工程领域。神经PDE求解器为传统数值方法提供了有前景的替代方案,但现有方法通常需要大量训练轨迹,而高保真PDE数据生成成本高昂。在数据受限时,其性能显著下降,暴露了数据效率低下问题。核心原因在于PDE动态蕴含强结构归纳偏置,但未被神经架构显式编码,导致模型需从数据中学习基本物理结构。尤其表现为对未见源项泛化能力差。本文基于格林函数理论——PDE理论基石——提出一种结构化归纳偏置的离散格林网络(DGNet)。核心思想是将格林函数转化为基于图的离散形式,并将叠加原理嵌入混合物理-神经架构,减轻从数据中学习物理先验的负担,从而提升样本效率。在多种时空PDE场景中,DGNet仅用数十条训练轨迹即实现最先进精度,并表现出对未见源项的稳健零样本泛化能力,验证了其高效结构设计的有效性。
原文摘要 · Abstract (English)
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.
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