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

TL;DR

The paper tackles exact enforcement of Dirichlet boundary conditions in physics-informed neural networks on convex polygonal domains by introducing a Wachspress-based transfinite interpolant as a lifting operator. The neural trial function is formed as $u_\theta^{\mathrm{TFI}} = g(\bm{\lambda}) + N_\theta(\bm{\lambda};\theta) - \mathscr{L}[N_\theta(\bm{\lambda};\theta)]$, ensuring the boundary values are met exactly and the Laplacian remains bounded, addressing the unbounded-Laplacian issue associated with approximate distance functions. Wachspress coordinates serve as a geometric feature map $\bm{\lambda}: \bar{P} \to [0,1]^n$, encoding boundary edges and enabling a framework for solving PDEs on parametrized convex geometries. The method is validated through forward, inverse, and parametrized Poisson problems on squares, quadrilaterals, pentagons, and a parametrized quadrilateral family, demonstrating high accuracy and training stability relative to ADF-based approaches. This work broadens the applicability of PINNs and deep Ritz methods to polygonal domains with exact BC enforcement, potentially informing geometry-aware neural operators and future extensions to polyhedra and nonconvex domains.

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, $\boldsymbolλ : \bar P \to [0,1]^n$, serve as a geometric feature map for the neural network: $\boldsymbolλ$ 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.

Figures (21)

• Figure 1: Computation of $\phi$ and $\nabla^2\phi$ over the unit square using R-equivalence with order of normalization $m$ = 1 (see Sukumar:2022:EIB). (a) $\phi$; and (b) $| \nabla^2 \phi |$. Note that $| \nabla^2 \phi | \to \infty$ at the vertices.
• Figure 2: One-dimensional interval.
• Figure 3: Boundary functions for transfinite interpolation using Wachspress coordinates. (a) Square; and (b) Quadrilateral.
• Figure 4: Boundary functions for transfinite interpolation using Wachspress coordinates on a pentagon.
• Figure 5: Computation of Wachspress coordinates on a polygon ($n = 4$).