Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's

Abstract

The present work is focused on exploring convergence of Physics-informed Neural Networks (PINNs) when applied to a specific class of second-order fully nonlinear Partial Differential Equations (PDEs). It is well-known that as the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We show that such sequence converges to a unique viscosity solution of a certain class of second-order fully nonlinear PDE's, provided the latter satisfies the comparison principle in the viscosity sense.

arXiv comment15 pages; Keywords: Physics Informed Neural Networks, Convergence, Viscosity Solutions, Differential Equations. arXiv admin note: substantial text overlap with arXiv:2004.01806 by other authors