Deep NURBS -- Admissible Physics-informed Neural Networks

TL;DR

The paper tackles the challenge of solving PDEs on complex geometries with Dirichlet boundaries using physics-informed neural networks. It introduces Deep NURBS, a method that strong-enforces boundary conditions by multiplying a neural network with an admissible NURBS-based field, and employs an IsoGeometric Analysis-inspired sampling strategy to realize automatic importance sampling. The approach yields fast convergence and high accuracy with shallow networks across non-Lipschitz domains (corner singularities, annular geometries, and perforated shapes), outperforming traditional PINN variants like Deep Ritz in several cases. This framework has potential to extend physics-informed learning to high-dimensional, geometry-rich variational problems with improved efficiency and robustness.

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.

Figures (13)

• Figure 1: Exemplification of Deep NURBS' method. From left to right: Admissible NURBS (ansatz) $\varphi(\boldsymbol{x})$, Neural Network $NN(\boldsymbol{x})$, and the product of the two first, that is, Admissible Neural Network $NN(\boldsymbol{x})\varphi(\boldsymbol{x})$, for the homogeneous Dirichlet case.
• Figure 2: Deep NURBS method illustrated. Integration of a Neural Network with NURBS functions to generate admissible Neural Networks.
• Figure 3: Parametric sampling.
• Figure 4: NURBS parameterization for the homogeneous Dirichlet boundary condition in $\mathcal{D}$ for problem \ref{['sec:exple1']}. NURBS: degree $p_1=p_2=2$, knot vectors: $\Xi^1=\{0,0,0,0.25,0.5,0.75,1,1,1\}$ and $\Xi^2=\{0,0,0,0.5,1,1,1\}$.
• Figure 5: Solution of the Poisson equation for problem \ref{['sec:exple1']}. We use the admissible field $(\phi,\boldsymbol{\zeta}=\boldsymbol{0})\in(\mathcal{P},\mathcal{Z})$ (a) to predict the solution using one (b) and two (c) hidden layers Deep NURBS solver. These results are to be compared to the exact solution (d) calculated using FEM