Amortizing Physics-Informed Neural Solvers via Graph Hypernetworks

2026-09-17

AI Summary

该研究通过图超网络生成初始化元训练因子化物理信息神经网络，以解决相关偏微分方程的求解问题，并在不同条件下验证了方法的有效性。

Abstract

Amortizing physics-informed neural networks (PINNs) across related PDEs requires describing each equation to a reusable solver. Coefficient vectors encode numerical parameters in predefined slots, leaving operator and cross-field assignments implicit. We make these relationships explicit in an operator graph, with nodes for fields, derivatives, terms, and residuals and coefficients retained as term attributes. A graph hypernetwork generates diagonal codes that initialize a meta-trained factorized PINN for each target equation. Meta-training and target-specific adaptation use governing equations and prescribed conditions without solution labels. We compare coefficient-vector, DeepSets-based term-set, and graph conditioning by solution accuracy within a fixed adaptation budget. In scalar convection-diffusion-reaction problems, both term-based descriptors improve high-reaction accuracy, with similar performance. In two-field Fisher-KPP, meta-training sees uncoupled and one-way systems; after 3,000 adaptation steps on unseen two-way coupling, the graph's mean final error is 35.7% below the term set and 67.7% below the coefficient vector. In a fixed-structure capacitively coupled plasma model, the coefficient vector performs best. These results support extending coefficient conditioning with explicit equation relationships for physics-based solver adaptation.

Problem

• Physics-Informed Neural Networks
• Graph Hypernetworks
• Partial Differential Equations
• Amortization
• Equation Relationships

Innovation

• Graph Hypernetworks
• Physics-Informed Neural Networks (PINNs)
• Operator Graph
• Meta-Training
• Adaptation