用神经网络解复杂设计模型的偏微分方程，精度媲美传统有限元。
Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domains
- 分块神经网络在参数域上求解，支持复杂几何建模。
- 边界与界面条件强约束，结果连续性好，误差小。
- 适合工程师用CAD模型直接求解物理场问题。
本文提出一种结合物理信息神经网络与多块等几何分析的计算框架，用于在复杂计算机辅助设计（CAD）几何体上求解偏微分方程。方法采用基于等几何分析参考域的局部神经网络，通过自定义输出层实现狄利克雷边界条件的强施加；利用专用接口神经网络确保非均匀有理B样条（NURBS）块之间的解连续性。训练基于变分框架，通过最小化弱形式导出的能量泛函完成。在两个高难度且具工程意义的应用案例中验证了方法的有效性：二维四极磁铁磁静力学模型与三维非线性固体力学及接触力学机械夹持器模型。结果与高保真有限元求解器获得的参考解高度一致，表明该神经求解器在给定CAD模型条件下解决复杂工程问题的潜力。
原文摘要 · Abstract (English)
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.
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