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.

翻译：本文提出了一种基于Wachspress坐标的凸多边形区域超越有限公式，用于在物理信息神经网络中精确施加Dirichlet边界条件。该方法借鉴了几何设计领域的先进成果，如凸区域上的混合函数与超越有限插值技术。对于给定的Dirichlet边界函数$\mathcal{B}$，其超越有限插值函数$g : \bar P \to C^0(\bar P)$能够将二维多边形区域边界上的函数$\textit{提升}$至区域内部。试函数被构造为神经网络输出与其边界限制在区域内部延拓值之差，并叠加$g$函数。这保证了试函数在深度Ritz方法中的运动学容许性。超越有限公式采用$n$边形的Wachspress坐标，将矩形上的双线性Coons超越有限插值推广至凸多边形情形。该神经网络试函数具有有界拉普拉斯算子，从而克服了先前采用近似距离函数精确施加Dirichlet边界条件时存在的局限性。对于点$\boldsymbol{x} \in \bar{P}$，Wachspress坐标$\boldsymbolλ : \bar P \to [0,1]^n$可作为神经网络的几何特征映射：$\boldsymbolλ$编码了多边形区域的边界边信息。这为使用神经网络求解参数化凸几何问题提供了理论框架。本文通过正问题、反问题及参数化几何的Poisson边值问题，评估了物理信息神经网络与深度Ritz方法的计算精度。