A Physics-Informed Neural Network with Conservation Laws for Solving KdV Equation

A Physics-informed neural network with conservation laws for solving Korteweg-de Vries equation is designed. By incorporating conservation laws into the loss function , initial condition into the network and minimizing the loss, the neural network parameters can be obtained. This neural network represents an approximate solution to the KdV equation with the initial conditions. To verify the effectiveness of the algorithm, numerical experiments were conducted using the one-soliton solution and the two-soliton solution of the KdV equation as initial conditions. The numerical results indicate that the PINN with conservation law algorithm has higher computational accuracy, has excellent generalization capabilities, has a smaller computational workload, and can maintain momentum and energy conservation better, which can maintain the waveform from deforming as time increases, compared to the PINN algorithm