A Wachspress-based transfinite formulation for exactly enforcing Dirichlet boundary conditions on convex polygonal domains in physics-informed neural networks

TL;DR

This paper introduces a Wachspress-based transfinite formulation for exactly enforcing Dirichlet boundary conditions in physics-informed neural networks on convex polygonal domains, improving boundary condition accuracy.

Contribution

It develops a novel Wachspress coordinate-based transfinite interpolation method that ensures exact Dirichlet boundary conditions in neural network solutions on convex polygons.

Findings

Achieves exact Dirichlet boundary enforcement in PINNs.

Demonstrates improved accuracy on linear and nonlinear problems.

Validates approach on inverse and parametrized boundary-value problems.

Abstract

In this paper, we present a Wachspress-based transfinite formulation on convex polygonal domains for exact enforcement of Dirichlet boundary conditions in physics-informed neural networks. This approach leverages prior advances in geometric design such as blending functions and transfinite interpolation over convex domains. For prescribed Dirichlet boundary function $\mathcal{B}$, the transfinite interpolant of $\mathcal{B}$, $g : \bar P \to C^0(\bar P)$, $\textit{lifts}$ functions from the boundary of a two-dimensional polygonal domain to its interior. The transfinite trial function is expressed as the difference between the neural network's output and the extension of its boundary restriction into the interior of the domain, with $g$ added to it. This ensures kinematic admissibility of the trial function in the deep Ritz method. Wachspress coordinates for an $n$-gon are used in the transfinite formula, which generalizes bilinear Coons transfinite interpolation on rectangles to convex polygons. Since Wachspress coordinates are smooth, the neural network trial function has a bounded Laplacian, thereby overcoming a limitation in a previous contribution where approximate distance functions were used to exactly enforce Dirichlet boundary conditions. For a point $\boldsymbol{x} \in \bar{P}$, Wachspress coordinates, $\boldsymbol{\lambda} : \bar P \to [0,1]^n$, serve as a geometric feature map for the neural network: $\boldsymbol{\lambda}$ encodes the boundary edges of the polygonal domain. This offers a framework for solving problems on parametrized convex geometries using neural networks. The accuracy of physics-informed neural networks is successfully assessed on forward problems (linear and nonlinear), an inverse heat conduction problem, and a parametrized geometric Poisson boundary-value problem.