Data-free neural PDE solvers based on Graph Neural Networks and weak forms

2026-07-30

AI Summary

This work addresses the limited generalization of conventional physics-informed neural networks (PINNs) for solving partial differential equations, which typically rely on abundant high-fidelity synthetic data. The authors propose a data-free, physics-driven graph neural network solver that uniquely integrates geometric deep learning with the variational weak formulation. By leveraging finite element shape function gradients—instead of automatic differentiation—to compute residuals and incorporating geometric inductive biases, the method generalizes to arbitrary unseen geometries and loading conditions without any training data. At inference time, it supports residual-driven adaptive mesh refinement. Experiments demonstrate that the approach achieves residual errors below 1% in previously unobserved scenarios, scales effectively to large-scale complex geometries, and eliminates the substantial costs associated with data generation and storage.

Abstract

We present a physics-informed, data-free neural solver for partial differential equations, built on a graph neural network architecture that utilises message passing. By relying on the weak form of the problem, we use gradients of finite-element shape functions (which are therefore polynomials) rather than automatic differentiation operators to compute the residuals of the equation from the displacements predicted by the network itself. Our approach generalises to previously unseen load cases and geometries, achieving easily convergence errors in the residuals of less than 1% and being capable of scaling up to models of considerable size and arbitrary geometries. To ensure compliance with the laws of physics and provide guarantees regarding the inference, it is possible to use the residual itself as an error indicator for the inference, and thus perform a refinement at the testing stage if the residual tolerance set in advance by the user is not met. Examples are provided to demonstrate the performance of the proposed method. This results in a method that avoids the costly process of obtaining, curating and storing high-fidelity synthetic data for training the neural network. Whilst this is not unique to our method, it is the first time it has been combined with a geometric machine learning technique capable of providing the necessary geometric bias to overcome the well-known difficulties of physics-informed neural networks.

Problem

• data-free
• neural PDE solvers
• graph neural networks
• weak form
• physics-informed

Innovation

• data-free
• graph neural networks
• weak form
• physics-informed
• geometric machine learning