Data-free neural PDE solvers based on Graph Neural Networks and weak forms
arXiv:2607.27901
The paper introduces a neural network that solves partial differential equations without any training data by using a graph neural network and the weak form of the equations, computing residuals with finite‑element shape function gradients and providing a built‑in error indicator for inference.
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.