Multi-benchmark adaptive sampling physics-informed neural network for complex and coupled equations

This paper introduces an innovative Multi-Benchmark Adaptive Sampling Physics-Informed Neural Network (MBAS-PINN) method, aimed at effectively addressing the challenges posed by complex-value and coupled equations. This method combines the advantages of PINN, which embeds physical information into the loss function of the neural network and leverages the approximation capabilities of the neural network to learn solution of equations. The introduced adaptive sampling strategy enables intelligent residual point selection by accounting for multiple factors, thereby enhancing solution accuracy for solving complex and coupled partial differential equations (PDEs). Specifically, the proposed adaptive sampling method first defines multiple benchmarks, which represent different characteristics or critical regions of solution. With two kinds of training strategies, the distribution of residual points is dynamically adjusted based on these benchmarks. In this way, the neural network can simultaneously focus more on the regions that have a greater impact on the real part and the imaginary part of solution, thereby accelerating convergence and improving the accuracy. In addition, we develop the neural tangent kernel of PINN for solving complex PDEs. To validate the effectiveness of this method, several experiments are conducted in some cases, including the nonlinear Schrödinger equation, the Hirota equation, and the Yajima-Oikawa system. Experimental results demonstrate that the MBAS-PINN method has achieved remarkable ability. In summary, this method provides a new and effective approach for solving complex-value and coupled PDEs.