A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation

TL;DR

This paper introduces a deep learning framework for solving PDEs on complex CAD geometries, combining variational neural networks with importance sampling and domain decomposition to improve accuracy in electric machine simulations.

Contribution

It develops a novel neural solver that integrates importance sampling and domain decomposition for PDEs on CAD geometries, enhancing solution accuracy and applicability to real-world engineering problems.

Findings

Neural solver achieves higher accuracy than traditional methods.

Effectiveness demonstrated on electric machine simulation.

Improved handling of multi-patch domains with discontinuities.

Abstract

This work presents a deep learning-based framework for the solution of partial differential equations on complex computational domains described with computer-aided design tools. To account for the underlying distribution of the training data caused by spline-based projections from the reference to the physical domain, a variational neural solver equipped with an importance sampling scheme is developed, such that the loss function based on the discretized energy functional obtained after the weak formulation is modified according to the sample distribution. To tackle multi-patch domains possibly leading to solution discontinuities, the variational neural solver is additionally combined with a domain decomposition approach based on the Discontinuous Galerkin formulation. The proposed neural solver is verified on a toy problem and then applied to a real-world engineering test case, namely that of electric machine simulation. The numerical results show clearly that the neural solver produces physics-conforming solutions of significantly improved accuracy.