Solving the KdV Equation Using the PINN Method

Abstract

Traditional numerical methods such as the Finite Difference Method (FDM) often exhibit significant numerical dissipation and dispersion errors when solving the Korteweg-de Vries (KdV) equation, particularly in long-term simulations or complex wave interactions like double soliton collisions. To address these limitations, this study proposes an improved approach based on Physics-Informed Neural Networks (PINNs). By embedding the physical constraints of the KdV equation directly into the loss function, and employing a lightweight fully-connected neural network with four hidden layers and automatic differentiation, the method enhances both accuracy and generalization capability. Numerical experiments on single and double soliton cases demonstrate that the PINN method significantly outperforms FDM and conventional Artificial Neural Networks (ANNs). For instance, in the single soliton case, the maximum error of PINN is reduced to 8.36e-03, with an L2 error of 1.33e-02, while FDM results in larger errors due to numerical artifacts. In the double soliton collision scenario, PINN effectively captures the nonlinear interaction process with minimal error accumulation. Although slight discrepancies are observed near the wave crest-especially for the faster wave-the overall waveform and dynamic behavior are accurately preserved. The absolute error distribution further confirms the stability and precision of the PINN solution across the computational domain. These results underscore the critical role of physical constraints in improving the modeling of nonlinear wave phenomena. The proposed PINN framework offers a robust, efficient, and high-fidelity alternative for solving complex PDEs in scientific and engineering applications.