  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.
