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Description

Training neural networks to solve partial differential equations (no labeled data) is challenging due to the need to balance competing loss terms, such as those from the residuals of the governing equation and boundary conditions. In this talk, I will present a Wachspress-based transfinite formulation on convex polygonal domains to exactly enforce Dirichlet boundary conditions in physics-informed neural networks (PINNs). For a prescribed Dirichlet boundary function B, the transfinite interpolant g extends B from the boundary of a two-dimensional polygonal domain to its interior. The neural network 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 leads to requiring just the PDE loss term in the training of neural networks.  Wachspress coordinates for a polygon, which are the generalization of barycentric coordinates from simplices to polytopes, are used in the transfinite formula. For a point in the polygon, Wachspress coordinates serve as a geometric feature map, and are used in the input layer of the neural network. This offers a framework for solving problems on parametrized convex geometries using neural networks. The accuracy of PINNs will be assessed on forward (linear and nonlinear) and inverse problems, and a parametrized geometric Poisson problem. This is joint work with Ritwick Roy at 3DS Simulia Corp.