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N. Sukumar, Professor, Computational Mechanics Department of Civil & Environmental Engineering, University of California, Davis
Exact Imposition of Boundary Conditions in Physics-Informed Neural Networks to Solve PDEs
In this presentation, I will first introduce a method based on distance fields to exactly impose boundary conditions in a PDE solver that is based on physics-informed neural networks (PINNs). In constructive solid geometry, R-functions provide a means to implicitly represent solids. We introduce an approach that uses R-functions to construct approximate distance fields in conjunction with transfinite interpolation to satisfy inhomogeneous Dirichlet, Neumann and Robin boundary conditions. Boundary-value problems that are posed in strong form and variational form (Ritz minimization) are considered. Instead of multiple (boundary and PDE) loss terms, only the PDE loss term is required, which improves the training of deep neural networks and better accuracy accrues for the solution of PDEs. A restriction of the method based on R-functions is that the Laplacian of the distance function blows up at the boundary vertices, which adversely affects training when collocation points are chosen to be located very close to the boundary vertices. I will present ongoing work that overcomes this limitation for Dirichlet boundary conditions over convex geometries, and relies on advances in blending functions and generalized barycentric maps over polygonal patches. The method based on R-functions is joint-work with Ankit Srivastava (IIT Chicago), and has been incorporated in the April 2022 release of NVIDIA Modulus (NVIDIA, Inc.).