Pre-Training Physics-Informed Neural Network with Mixed Sampling and Its Application in High-Dimensional Systems

Abstract

Recently, the physics-informed neural network shows remarkable ability in the context of solving the low-dimensional nonlinear partial differential equations. However, for some cases of high-dimensional systems, such technique may be time-consuming and inaccurate. In this paper, the authors put forward a pre-training physics-informed neural network with mixed sampling (pPINN) to address these issues. Just based on the initial and boundary conditions, the authors design the pre-training stage to filter out the set of the misfitting points, which is regarded as part of the training points in the next stage. The authors further take the parameters of the neural network in Stage 1 as the initialization in Stage 2. The advantage of the proposed approach is that it takes less time to transfer the valuable information from the first stage to the second one to improve the calculation accuracy, especially for the high-dimensional systems. To verify the performance of the pPINN algorithm, the authors first focus on the growing-and-decaying mode of line rogue wave in the Davey-Stewartson I equation. Another case is the accelerated motion of lump in the inhomogeneous Kadomtsev-Petviashvili equation, which admits a more complex evolution than the uniform equation. The exact solution provides a perfect sample for data experiments, and can also be used as a reference frame to identify the performance of the algorithm. The experiments confirm that the pPINN algorithm can improve the prediction accuracy and training efficiency well, and reduce the training time to a large extent for simulating nonlinear waves of high-dimensional equations.