Enhancing accuracy of physically informed neural networks for nonlinear Schrödinger equations through multi-view transfer learning

Abstract

In recent years, significant research efforts have been dedicated to developing solutions for nonlinear partial differential equations (PDEs) with applications in physics. Among these equations, the NonLinear Schrödinger (NLS) equation plays a pivotal role in modern nonlinear science, governing the behavior of optical solitons and serving as a crucial model. However, the practical application of these elegant PDE models is hindered by the challenge of high dimensionality and computational cost. To address these challenges, we introduce a novel approach to enhance the training accuracy of physically informed neural networks (PINNs) for the NLS equation. Leveraging the concept of multi-view transfer learning, we utilize a pre-trained NLS model that considers different initial–boundary conditions as multiple views, enabling exploration of their contributions to the overall understanding and solution of the NLS equation. Additionally, we investigate the influence of different loss functions on PINN training results and find that employing the Huber Loss function yields superior fitting outcomes compared to traditional PINNs that use mean squared error (MSE) loss. We demonstrate significant improvements in precision and accuracy over traditional PINNs. Extensive experiments validate the effectiveness of our proposed optimization techniques. This work paves a novel way for solving complex physical systems using data-driven approaches.