Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains

TL;DR

This paper establishes rigorous error bounds for Physics Informed Neural Networks approximating the generalized KdV equations on unbounded domains, addressing the challenges of oscillatory behavior and complex solutions.

Contribution

It adapts advanced oscillatory estimate techniques to the deep learning framework and provides the first rigorous approximation bounds for gKdV models using PINNs.

Findings

Rigorous error bounds for PINNs on unbounded domains.

Effective approximation of solitons and breathers.

Extension of oscillatory estimate techniques to neural network analysis.

Abstract

In this paper we study a rigorous setting for the numerical approximation via deep neural networks of the generalized Korteweg-de Vries (gKdV) model in one dimension, for subcritical and critical nonlinearities, and assuming that the domain is the unbounded real line. The fact that the model is posed on the real line makes the problem difficult from the point of view of learning techniques, since the setting required to model gKdV is structured on intricate oscillatory estimates dating from Kato, Bourgain and Kenig, Ponce and Vega, among others. Therefore, a first task is to adapt the setting of these techniques to the deep learning setting. We shall use a battery of Kenig-Ponce-Vega suitable norms and Physics Informed Neural Networks (PINNs) to describe this approximative scheme, proving rigorous bounds on the approximation for each critical and subcritical gKdV model. We shall use this results to provide clear approximation results in the case of several gKdV nonlinear patterns such as solitons, multi-solitons, breathers, among other solutions.