Error estimates for viscous Burgers' equation using deep learning method
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.
Cite
@article{arxiv.2502.19392,
title = {Error estimates for viscous Burgers' equation using deep learning method},
author = {Wasim Akram and Sagar Gautam and Deepanshu Verma and Manil T. Mohan},
journal= {arXiv preprint arXiv:2502.19392},
year = {2025}
}