Error estimates for viscous Burgers' equation using deep learning method

TL;DR

This paper develops error estimates and stability analysis for deep learning approximations of the viscous Burgers' equation, providing theoretical guarantees and numerical validation for both stationary and non-stationary cases.

Contribution

It introduces a rigorous framework for error estimation and stability analysis of deep neural network methods applied to the viscous Burgers' equation, including proofs of well-posedness and explicit error bounds.

Findings

Derived explicit error estimates in Lebesgue and Sobolev norms.

Proved local and global well-posedness of the problem.

Numerical results validate theoretical error bounds.

Abstract

The article focuses on error estimates as well as stability analysis of deep learning methods for stationary and non-stationary viscous Burgers equation in two and three dimensions. The local well-posedness of homogeneous boundary value problem for non-stationary viscous Burgers equation is established by using semigroup techniques and fixed point arguments. By considering a suitable approximate problem and deriving appropriate energy estimates, we prove the existence of a unique strong solution. Additionally, we extend our analysis to the global well-posedness of the non-homogeneous problem. For both the stationary and non-stationary cases, we derive explicit error estimates in suitable Lebesgue and Sobolev norms by optimizing a loss function in a Deep Neural Network approximation of the solution with fixed complexity. Finally, numerical results on prototype systems are presented to illustrate the derived error estimates.