Deep NURBS -- Admissible Physics-informed Neural Networks
This work addresses the challenge of enforcing boundary conditions in PINNs for PDEs with complex geometries, which is important for computational physics and engineering applications, representing a novel method rather than an incremental improvement.
The authors tackled the problem of solving partial differential equations (PDEs) with arbitrary geometries and strict Dirichlet boundary conditions by proposing Deep NURBS, a physics-informed neural network (PINN) method that combines admissible NURBS parametrizations with PINN solvers, achieving high convergence rates and desirable accuracy with only one hidden layer in most cases.
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.