  The numerical determination of solitary states is an important topic for such
research areas as Bose-Einstein condensates, nonlinear optics, plasma physics,
etc. In this paper, we propose a data-driven approach for identifying solitons
based on dynamical solutions of real-time differential equations. Our approach
combines a machine-learning architecture called the complex-valued neural
operator (CNO) with an energy-restricted gradient optimization. The former
serves as a generalization of the traditional neural operator to the complex
domain, and constructs a smooth mapping between the initial and final states;
the latter facilitates the search for solitons by constraining the energy
space. We concretely demonstrate this approach on the quasi-one-dimensional
Bose-Einstein condensate with homogeneous and inhomogeneous nonlinearities. Our
work offers a new idea for data-driven effective modeling and studies of
solitary waves in nonlinear physical systems.
