The nonlinear Schrödinger equation is a fundamental model for describing the propagation of wave packets in dispersive and nonlinear media. It is widely used in nonlinear optics, plasma physics, and other fields that involve complex wave dynamics. This paper proposes PI-CVNN, a physics-informed complex-valued neural network for solving the nonlinear Schrödinger equation. The proposed method directly models the optical envelope in the complex domain and incorporates the initial condition, boundary condition, and governing-equation residual into a unified physics-informed loss function. By using complex-valued weight matrices, PI-CVNN explicitly represents the coupling between the real and imaginary parts, thereby better characterizing the evolution of the complex envelope. Experimental results show that PI-CVNN effectively approximates the complex-valued dynamics of optical soliton propagation in fibers. This study provides a useful tool for modeling nonlinear effects in optical fiber communications and broadens the application potential of complex-valued neural networks in scientific computing.