Data-driven solution dynamics and parameters discovery for the weakly nonlocal Schrödinger equation with parabolic law nonlinearity and an external potential via PINNs deep learning
摘要
The weak nonlocal Schrödinger (WNS) equation plays a crucial role in describing complex physical phenomena, such as optical solitons in nonlocal nonlinear media. Therefore, studying its dynamic behavior and identifying its physical parameters have significant research value. This paper utilizes physics-informed neural networks (PINNs) to systematically investigate a WNS equation with parabolic law nonlinearity and an external potential, and realizes the data-driven solutions and parameter discovery of this equation. In this study, by constructing a novel loss function that includes equation residuals, initial-boundary value conditions, and observational data, the dynamic characteristics of various solutions such as dark solitons, bright solitons, and exponential solutions are successfully learned. The impact of different neural network architectures and activation functions on model performance is explored, revealing that the hyperbolic tangent and sine functions are best suited for learning soliton and exponential solutions, respectively. The research results show that this method can accurately learn the dynamic behavior for the solutions of the WNS equation with a relative error of