Adaptive sampling physical neural network based on conservation law for solving Korteweg–de Vries equation

Solving the Korteweg–de Vries (KdV) equation with strong nonlinearity and dispersion characteristics is a pivotal issue in the field of nonlinear dynamics. Traditional numerical methods are constrained by grid resolution and numerical stability and face bottlenecks in accuracy and efficiency when dealing with high-dimensional and multi-scale problems. Physics-Informed Neural Networks (PINNs) provide a novel approach to solving partial differential equations due to their mesh-free nature and capability of embedding physical constraints. However, the adaptive sampling strategy of PINNs is solely guided by equation residuals, which tends to result in the problem that the numerical solution satisfies the formal structure of the equation but violates the physical conservation laws. To address this deficiency, this paper proposes a residual adaptive distribution sampling method with conservation constraints (RAD-C), taking the three core conserved quantities mass, momentum, and energy of the Korteweg–de Vries equation as physical benchmarks and introducing a weight coefficient to optimize the sampling probability density function, achieving dual-focus sampling on regions with high equation residuals and large conservation deviations. The experimental results demonstrate that, compared to the traditional sampling method, the RAD-C method can maintain high precision even in complex scenarios with increased soliton amplitude and narrowed width with only marginal additional computational cost. The L2 relative error of the one-soliton simulation can reach 0.0090, and the error control effect is more significant in the two-soliton collision simulation, in which the convergence accuracy of the loss function stably reaches the order of 10−5.