# Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domains

arXiv: https://arxiv.org/abs/2509.25450 · HTML: https://mlsanity.com/paper/2509.25450/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 69 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Neural networks meet hyperelasticity: A guide to enforcing physics](https://mlsanity.com/paper/2302.02403/) (2023)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [NURBS-Diff: A Differentiable Programming Module for NURBS](https://mlsanity.com/paper/2104.14547/) (2021)
- [Physics-informed neural networks for solving parametric magnetostatic problems](https://mlsanity.com/paper/2202.04041/) (2022)
- [Deep NURBS -- Admissible Physics-informed Neural Networks](https://mlsanity.com/paper/2210.13900/) (2022)
- [A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation](https://mlsanity.com/paper/2111.09005/) (2021)
- [Multi-Patch Isogeometric Convolution Hierarchical Deep-learning Neural Network](https://mlsanity.com/paper/2406.03307/) (2024)
- [Torque Computation with the Isogeometric Mortar Method for the Simulation of Electric Machines](https://mlsanity.com/paper/2202.05771/) (2022)
- [Error Approximation and Bias Correction in Dynamic Problems using a Recurrent Neural Network/Finite Element Hybrid Model](https://mlsanity.com/paper/2307.02349/) (2023)

## Abstract (arXiv)

This work develops a computational framework that combines physics-informed neural networks with multi-patch isogeometric analysis to solve partial differential equations on complex computer-aided design geometries. The method utilizes patch-local neural networks that operate on the reference domain of isogeometric analysis. A custom output layer enables the strong imposition of Dirichlet boundary conditions. Solution conformity across interfaces between non-uniform rational B-spline patches is enforced using dedicated interface neural networks. Training is performed using the variational framework by minimizing the energy functional derived after the weak form of the partial differential equation. The effectiveness of the suggested method is demonstrated on two highly non-trivial and practically relevant use-cases, namely, a 2D magnetostatics model of a quadrupole magnet and a 3D nonlinear solid and contact mechanics model of a mechanical holder. The results show excellent agreement to reference solutions obtained with high-fidelity finite element solvers, thus highlighting the potential of the suggested neural solver to tackle complex engineering problems given the corresponding computer-aided design models.