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

TL;DR

This paper proves that Physics-Informed Neural Networks (PINNs) converge to the unique viscosity solution for a class of second-order fully nonlinear PDEs under certain conditions, advancing theoretical understanding of PINN reliability.

Contribution

It establishes the convergence of PINNs to viscosity solutions for fully nonlinear PDEs, a significant theoretical result in the field.

Findings

PINNs generate a sequence of neural networks as data increases.

The sequence converges to the viscosity solution under the comparison principle.

Provides theoretical validation for PINNs solving nonlinear PDEs.

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.