Interpretable Neural Network Quantum States for Solving the Steady States of the Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces an interpretable neural network approach to solve both ground and excited states of the nonlinear Schrödinger equation, enabling analysis of complex chaotic wave phenomena.

Contribution

It presents a novel neural network quantum state method with interpretable architectures for directly computing nonlinear Schrödinger equation solutions, including excited states.

Findings

Successfully computed excited states of NLSE

Demonstrated application to spatiotemporal chaos

Provided analytical approximations of wave solutions

Abstract

The nonlinear Schr\"odinger equation (NLSE) underpins nonlinear wave phenomena in optics, Bose-Einstein condensates, and plasma physics, but computing its excited states remains challenging due to nonlinearity-induced non-orthonormality. Traditional methods like imaginary time evolution work for ground states but fail for excited states. We propose a neural network quantum state (NNQS) approach, parameterizing wavefunctions with neural networks to directly minimize the energy functional, enabling computation of both ground and excited states. By designing compact, interpretable network architectures, we obtain analytical approximation of solutions. We apply the solutions to a case of spatiotemporal chaos in the NLSE, demonstrating its capability to study complex chaotic dynamics. This work establishes NNQS as a tool for bridging machine learning and theoretical studies of chaotic wave systems.