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

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.

💡 Research Summary

This paper introduces HWF-PIKAN, a novel Multi-Resolution Hybrid Wavelet-Fourier Physics-Informed Kolmogorov-Arnold Network, designed to overcome the limitations of existing physics-informed neural networks in solving complex partial differential equations (PDEs), particularly those involving multi-scale features and discontinuities.

The core innovation lies in a sophisticated input embedding strategy that combats spectral bias. Before being processed by the neural network core, each normalized input coordinate is mapped into a hybrid spectral feature space. This space combines two complementary representations: 1) Fourier Embedding: Uses deterministic integer-frequency sine and cosine functions to capture global, smooth variations across the domain. 2) Wavelet Embedding: Employs multi-scale Ricker (Mexican-hat) wavelets with decreasing scale parameters, enabling the model to detect localized, sharp features and abrupt changes at different resolutions. The concatenated hybrid feature vector is then passed to the network’s core.

The core architecture is based on the recently proposed Kolmogorov-Arnold Network (KAN), which implements the Kolmogorov-Arnold Representation Theorem. Unlike standard Multi-Layer Perceptrons (MLPs) that apply fixed activation functions at nodes, KANs place learnable univariate activation functions (composed of weighted B-spline bases and a SiLU function) on the edges (weights). This structure allows for a hierarchical decomposition of high-dimensional functions into simpler univariate transformations, offering greater efficiency and interpretability in function approximation.

The model is trained in a fully data-free manner. The loss function penalizes the residuals of the governing PDE, the initial conditions (ICs), and the boundary conditions (BCs), forcing the network to discover a solution that satisfies all physical constraints simultaneously.

The authors validate HWF-PIKAN through a two-stage experimental process. First, systematic benchmarks on 1D and 2D advection equations with both continuous and discontinuous initial conditions (e.g., square waves) demonstrate its superior accuracy and faster convergence compared to Vanilla PINN, Vanilla PIKAN, and PIKAN variants enhanced solely with either Fourier or wavelet features. The hybrid model excels at capturing both smooth propagation and sharp fronts without oscillatory artifacts.

Second, the framework is extended to solve the high-dimensional Collisionless Boltzmann Equation (CBE or Vlasov equation) in phase-space. This kinetic equation, fundamental to plasma physics and astrophysics, is notoriously challenging due to its high dimensionality and the development of fine-scale filamentation structures. The paper adopts a continuous-velocity formulation, treating velocity as a continuous input variable alongside space and time, thereby avoiding the expensive velocity-space discretization required by traditional mesh-based methods. In this high-dimensional setting, HWF-PIKAN successfully learns the phase-space density distribution, accurately resolving the complex filamentation patterns that emerge. It significantly outperforms all other compared models in this task.

In conclusion, this work highlights the power of multi-resolution spectral feature embeddings to advance physics-informed deep learning. HWF-PIKAN presents a robust, mesh-free framework capable of handling PDEs with diverse scale characteristics, from simple advection problems to complex, high-dimensional kinetic equations in phase-space. The publicly released code ensures reproducibility and provides a valuable tool for the computational science community.