Error estimates for physics informed neural networks approximating the Navier-Stokes equations

TL;DR

This paper establishes rigorous error bounds for physics-informed neural networks approximating the Navier-Stokes equations, linking residuals to training error, network size, and quadrature points, supported by numerical experiments.

Contribution

It provides the first rigorous error estimates for physics-informed neural networks applied to Navier-Stokes equations, connecting residuals to training and network parameters.

Findings

Residuals can be made arbitrarily small with tanh networks.

Total error estimates depend on training error, network size, and quadrature points.

Numerical experiments validate the theoretical bounds.

Abstract

We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.