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

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