The numerical determination of solitary states is an important topic for such research areas as Bose-Einstein condensates, nonlinear optics, plasma physics, and so on. 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 CNO 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 energy-restricted optimization 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 an idea for data-driven effective modeling and studies of solitary waves in nonlinear physical systems.