HWF-PIKAN: A Multi-Resolution Hybrid Wavelet-Fourier Physics-Informed Kolmogorov-Arnold Network for solving Collisionless Boltzmann Equation

Abstract: Physics-Informed Neural Networks (PINNs) and more recently Physics-Informed Kolmogorov-Arnold Networks (PIKANs) have emerged as promising approaches for solving partial differential equations (PDEs) without reliance on extensive labeled data. In this work, we propose a novel multi-resolution Hybrid Wavelet-Fourier-Enhanced Physics-Informed Kolmogorov-Arnold Network (HWF-PIKAN) for solving advection problems based on collisionless Boltzmann equation (CBE) with both continuous and discontinuous initial conditions. To validate the effectiveness of the proposed model, we conduct systematic benchmarks on classical advection equations in one and two dimensions. These tests demonstrate the model's ability to accurately capture smooth and abrupt features. We then extend the application of HWF-PIKAN to the high-dimensional phase-space setting by solving the CBE in a continuous-velocity manner. This leverages the Hamiltonian concept of phase-space dynamics to model the statistical behavior of particles in a collisionless system, where advection governs the evolution of a probability distribution function or number density. Comparative analysis against Vanilla PINN, Vanilla PIKAN, as well as Fourier-enhanced and Wavelet-enhanced PIKAN variants, shows that the proposed hybrid model significantly improves solution accuracy and convergence speed. This study highlights the power of multi-resolution spectral feature embeddings in advancing physics-informed deep learning frameworks for complex kinetic equations in both space-time and phase-space.