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

2025-09-29

AI Summary

Solving partial differential equations (PDEs) on complex CAD geometries remains challenging due to difficulties in mesh generation, enforcing strong boundary conditions, and ensuring inter-patch continuity. This work proposes a variational framework integrating multi-patch isogeometric analysis with physics-informed neural networks (PINNs). Leveraging NURBS-based multi-patch representations, it constructs patch-local PINNs and introduces dedicated interface networks to enforce C⁰ continuity across patches. A variational energy minimization loss, combined with a Dirichlet-constrained output layer, enables end-to-end solutions that are geometrically faithful, boundary- and interface-consistent, and highly accurate. To our knowledge, this is the first approach unifying patch-local neural networks with variational PINNs to systematically address strong Dirichlet enforcement and multi-patch coordination in realistic engineering CAD models. Numerical experiments—including 2D static magnetic fields in a quadrupole magnet and 3D nonlinear contact mechanics—demonstrate excellent agreement with high-fidelity finite element benchmarks, validating the method’s efficacy and generalizability in practical engineering applications.

Technology Category

Constraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationComputer Vision: Generative Adversarial Networks (GANs) for VisionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applicationsWeb Mining and Content Analysis: Large pretrained models with web data

Abstract

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.

Problem

• Solving PDEs on complex CAD geometries using neural networks
• Enforcing boundary conditions across multi-patch NURBS interfaces
• Demonstrating accuracy through electromagnetic and mechanical case studies

Innovation

• Combines neural networks with multi-patch isogeometric analysis
• Uses patch-local networks on reference domains
• Enforces solution conformity with interface networks