Error Estimates for Deep Learning Methods in Fluid Dynamics

TL;DR

This paper derives explicit error estimates and stability results for deep learning methods applied to fluid dynamics problems, specifically the Navier-Stokes equations, providing theoretical guarantees for DNN approximations.

Contribution

It offers the first rigorous error and stability analysis for deep neural network solutions to certain partial differential equations in fluid dynamics.

Findings

Explicit error bounds for DNN approximations of Navier-Stokes solutions

Stability analysis of deep learning methods in fluid dynamics

Theoretical validation of DNN-based PDE solvers

Abstract

In this study, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier-Stokes equations. In particular, we obtain explicit error estimates (in suitable norms) for the solution computed by optimizing a loss function in a Deep Neural Network (DNN) approximation of the solution, with a fixed complexity.