Identification of moment equations via data-driven approaches in nonlinear schrodinger models

TL;DR

This paper introduces a data-driven SINDy-based method to identify and analyze the evolution of moment quantities in nonlinear Schrödinger models, including cases without closed-form solutions.

Contribution

It applies SINDy to capture moment dynamics in nonlinear Schrödinger equations, exploring coordinate transformations and extending to non-closed systems.

Findings

Successfully identified moment evolution in known ODE systems

Explored different SINDy library choices for better modeling

Extended approach to systems lacking closed-form moment equations

Abstract

The moment quantities associated with the nonlinear Schrodinger equation offer important insights towards the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities is amenable to both analytical and numerical treatments. In this paper we present a data-driven approach associated with the Sparse Identification of Nonlinear Dynamics (SINDy) to numerically capture the evolution behaviors of such moment quantities. Our method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.