论文标题
深度NURBS-可接受的物理知识神经网络
Deep NURBS -- Admissible Physics-informed Neural Networks
论文作者
论文摘要
在这项研究中，我们提出了一种针对物理信息的神经网络（PINN）的新数值方案，该方案在任意几何形状的情况下，在严格执行DIRICHLET边界条件的同时，为部分微分方程（PDE）提供了精确且廉价的解决方案。所提出的方法结合了可允许的NURBS参数化，以用PINN求解器定义物理域和Dirichlet边界条件。在这个新颖的Deep Nurbs框架中，基本的边界条件会自动满足。在考虑任意几何形状（包括非lipschitz域）时，我们使用二维椭圆PDE验证了我们的新方法。与经典的PINN求解器相比，Deep Nurbs估计器对于所有研究的问题的收敛速率都非常高。此外，对于大多数研究的PDE，仅使用一个隐藏的神经网络层就实现了理想的精度。这种新颖的方法被认为是通过允许更现实的物理信息统计学习来解决基于PDE的变异问题，为高维问题铺平了解决方案的道路。
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.
