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

2026-01-05

arXiv.org

AI Summary

This work addresses the challenge of accurately enforcing Dirichlet boundary conditions in physics-informed neural networks (PINNs) over convex polygonal domains. The authors propose a transfinite interpolation method based on Wachspress coordinates to exactly extend boundary functions into the interior, thereby constructing trial neural network functions that strictly satisfy the prescribed boundary conditions and possess bounded Laplacians. This approach represents the first integration of Wachspress coordinates with transfinite interpolation, overcoming limitations inherent in conventional approximate distance functions and offering applicability to arbitrary convex polygonal geometries. Numerical experiments on forward, inverse, and parametric Poisson boundary value problems demonstrate that the method achieves high accuracy, exhibits strong generalization capabilities, and enables a unified solution framework for parameterized convex domains.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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 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. 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 and deep Ritz is assessed on forward, inverse, and parametrized geometric Poisson boundary-value problems.

Problem

• Dirichlet boundary conditions
• physics-informed neural networks
• convex polygonal domains
• exact enforcement
• boundary-value problems

Innovation

• Wachspress coordinates
• transfinite interpolation
• Dirichlet boundary conditions
• physics-informed neural networks
• convex polygonal domains