Exact Boundary Enforcement Along Implicit Geometries for Physics-Informed, Deep Learning Problems in Continuum Mechanics

Abstract

Solutions to well-posed problems in continuum mechanics are continuously dependent upon prescribed boundary conditions. Because of this, variations in the enforcement of boundary data can impact the reliability of inversion techniques that rely on efficient and accurate forward models. To this end, it is necessary to understand how specific boundary implementation techniques can affect the performance of a given forward model. Our work focuses on the impact that key modeling decisions have on physics-informed neural network (PINN) solutions for initial boundary value problems in continuum mechanics. By interpolating boundary data over implicit boundary representations, we measure the performance of a physics-informed neural network across different configurations of soft and hard boundary enforcement. We target the problem of elastodynamic plane-strain and present a method of hard-enforcement of traction conditions over arbitrary, implicitly-defined, domain boundaries considering both first and second order formulations of the governing equations. We show that PINNs achieve a higher relative accuracy when solving the first-order plane strain problem and we observe a tradeoff between the final relative error and the total run time to complete training. This tradeoff is characterized by the number of hard and soft boundaries where, in the extremes, all soft-enforcement results in greater accuracy with a longer run time, while all hard-enforcement leads to lesser accuracy and a shorter run time.