Deep NURBS -- Admissible Physics-informed Neural Networks

TL;DR

Deep NURBS introduces a novel physics-informed neural network framework that accurately solves PDEs on complex geometries by integrating NURBS parametrizations, automatically satisfying boundary conditions, and achieving high convergence with minimal network complexity.

Contribution

It combines NURBS-based domain parametrization with PINNs to automatically enforce boundary conditions and improve convergence on arbitrary geometries.

Findings

High convergence rate across studied PDEs.

Achieved accurate solutions with only one hidden neural network layer.

Effective on non-Lipschitz and complex geometries.

Abstract

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.