Graph hypernetworks improve adaptation of neural solvers for related physics problems
Amortizing Physics-Informed Neural Solvers via Graph Hypernetworks
Machine Learning
Summary
Solving physics problems with math equations called partial differential equations (PDEs) can be slow when done many times for similar problems. The authors improved neural networks designed to solve these equations by representing the math in a graph that clearly shows how parts of the equation relate. This lets their system learn from many related equations and quickly adapt to new ones, improving accuracy without needing example solutions. They tested this idea on different physics problems and found the graph-based approach often worked better than simpler ways to describe the equations.
What this means in practice
- •For computational engineers: Accelerate solving related physics PDEs by quickly adapting pretrained neural solvers with explicit graph-based equation descriptions.
- •For simulation software developers: Integrate graph hypernetwork conditioning to enhance solver accuracy on multi-field coupled PDE models without labeled data.
Authors
Cheng Jing, Abhishek Verma, Kallol Bera, Yixuan He, Kookjin Lee
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.