Interpretable neural network quantum states for solving the steady states of the nonlinear Schrödinger equation

The nonlinear Schrödinger 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, such as 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.