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

arXiv: https://arxiv.org/abs/2111.09005 · HTML: https://mlsanity.com/paper/2111.09005/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2021.

## Built on by (most cited first, 2 of 2)

- [A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials](https://mlsanity.com/paper/2605.31231/) (2026)
- [Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domains](https://mlsanity.com/paper/2509.25450/) (2025)

## Abstract (arXiv)

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.