Error bounds for Physics Informed Neural Networks in Nonlinear Schr\"odinger equations placed on unbounded domains

TL;DR

This paper develops error bounds for Physics-Informed Neural Networks approximating solutions to the nonlinear Schrödinger equation on unbounded domains, extending previous bounded domain results with theoretical guarantees and numerical validation.

Contribution

Introduces a new PINNs method for unbounded domains and provides rigorous error bounds based on energy and Strichartz norms, with applications to solitons and traveling waves.

Findings

Established error bounds for PINNs in unbounded NLS problems

Validated the approach with numerical experiments on solitons and breathers

Extended PINNs applicability beyond bounded domains

Abstract

We consider the subcritical nonlinear Schr\"odinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.