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, $\boldsymbolλ : \bar P \to [0,1]^n$, 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 is successfully assessed on forward problems (linear and nonlinear), an inverse heat conduction problem, and a parametrized geometric Poisson boundary-value problem.

翻译：本文提出了一种基于Wachspress坐标的凸多边形区域跨有限元公式，用于在物理信息神经网络中精确施加狄利克雷边界条件。该方法利用了几何设计领域的先前进展，如凸区域上的混合函数和跨有限元插值。对于给定的狄利克雷边界函数 $\mathcal{B}$，其跨有限元插值 $g : \bar P \to C^0(\bar P)$ 将函数从二维多边形区域的边界提升到其内部。跨有限元试验函数表示为神经网络输出与其边界限制在区域内部扩展之间的差值，并加上 $g$。这确保了深度Ritz方法中试验函数的运动学可容许性。在跨有限元公式中使用了 $n$ 边形的Wachspress坐标，将矩形上的双线性Coons跨有限元插值推广到了凸多边形。由于Wachspress坐标是光滑的，神经网络试验函数具有有界拉普拉斯算子，从而克服了先前方法中使用近似距离函数精确施加狄利克雷边界条件的局限性。对于点 $\boldsymbol{x} \in \bar{P}$，Wachspress坐标 $\boldsymbolλ : \bar P \to [0,1]^n$ 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。这为使用神经网络求解参数化凸几何问题提供了一个框架。物理信息神经网络的精度在正向问题（线性和非线性）、一个逆热传导问题以及一个参数化几何泊松边值问题上得到了成功评估。