<p>In this paper, we focus on a nonlinear Schrödinger (NLS) equation, constructed by introducing Brownian motion and Gaussian white noise terms into the standard NLS equation. First, the bilinear form of this equation is derived using the Hirota method. Based on the perturbation method, first- and second-order soliton solutions are obtained. By adjusting parameters, various forms of analytical solutions are acquired, and the influence of parameter variations on the solutions is analyzed with the aid of soliton graphs. When the perturbation generates bright solitons, one soliton exhibits interactions similar to those of two solitons, while two solitons demonstrates behavior akin to three solitons. If the perturbation produces dark solitons or periodic backgrounds, it can excite the generation of breathers. Furthermore, different perturbations cause breathers to exhibit distinct interaction patterns. Furthermore, we introduce the Physics-Informed Neural Networks from deep learning to predict soliton solutions, obtaining highly accurate predicted results. Then, the bilinear Neural Network method is employed to achieve high-precision approximation of numerical solutions for nonlinear partial differential equations through parameter optimization, yielding bright solitons, dark solitons, and periodic solutions. Finally, we employ the linear stability analysis method to investigate the stability of the solitons.</p>