In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solution for partial differential equations (PDEs) in case of arbitrary geometries while strictly enforcing Dirichlet boundary conditions. The proposed approach combines admissible NURBS parametrizations required to define the physical domain and the Dirichlet boundary conditions with a PINN solver. The fundamental boundary conditions are automatically satisfied in this novel Deep NURBS framework. We verified our new approach using two-dimensional elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the classical PINN solver, the Deep NURBS estimator has a remarkably high convergence rate for all the studied problems. Moreover, a desirable accuracy was realized for most of the studied PDEs using only one hidden layer of neural networks. This novel approach is considered to pave the way for more effective solutions for high-dimensional problems by allowing for more realistic physics-informed statistical learning to solve PDE-based variational problems.
翻译:在这项研究中,我们为物理知情神经网络提出了一个新的数字计划,在任意的地貌特征下,在严格执行Drichlet边界条件的同时,为部分差异方程(PDE)提供精确和廉价的解决方案。拟议方法将界定物理域和Drichlet边界条件所需的可接受 NURBS 配有PINN 求解器的可接受 NURBS 配方。在这个新型的深海NURBS框架中,基本边界条件自动得到满足。我们在考虑包括非Lipschitz 域在内的任意的地貌特征时,用二维椭圆形 PDE 验证了我们的新办法。与古典的 PINNS 求解器相比,深 NURBS 估计器对所有研究的问题具有极高的趋同率。此外,大多数已研究的PDE仅使用一个隐藏的神经网络层实现了理想的准确性。这种新办法被认为为更有效地解决高维度问题铺平道路,通过更现实的物理知情的统计学习解决基于PDE的变异性问题。