Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Abstract

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct $\varphi$, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as $\varphi$ multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. We present numerical solutions for linear and nonlinear boundary-value problems over domains with affine and curved boundaries. Benchmark problems in 1D for linear elasticity, advection-diffusion, and beam bending; and in 2D for the Poisson equation, biharmonic equation, and the nonlinear Eikonal equation are considered. The approach extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

50 pages, 46 figures

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• SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
• A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions
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• NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework
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• Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains
• Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learnings