Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries

Abstract

Since their advent nearly a decade ago, physics-informed neural networks (PINNs) have been studied extensively as a novel technique for solving forward and inverse problems in physics and engineering. The neural network discretization of the solution field is naturally adaptive and avoids meshing the computational domain, which can both improve the accuracy of the numerical solution and streamline implementation. However, there have been limited studies of PINNs on complex three-dimensional geometries, as the lack of mesh and the reliance on the strong form of the partial differential equation (PDE) make boundary condition (BC) enforcement challenging. Techniques to enforce BCs with PINNs have proliferated in the literature, but a comprehensive side-by-side comparison of these techniques and a study of their efficacy on geometrically complex three-dimensional test problems are lacking. In this work, we i) systematically compare BC enforcement techniques for PINNs, ii) propose a general solution framework for arbitrary three-dimensional geometries, and iii) verify the methodology on three-dimensional, linear and nonlinear test problems with combinations of Dirichlet, Neumann, and Robin boundaries. Our approach is agnostic to the underlying PDE, the geometry of the computational domain, and the nature of the BCs, while requiring minimal hyperparameter tuning. This work represents a step in the direction of establishing PINNs as a mature numerical method, capable of competing head-to-head with incumbents such as the finite element method.