Solving the BGK Model and Boltzmann equation by Fourier Neural Operator with conservative constraints
Boyun Hu, Kunlun Qi
TL;DR
The paper tackles the computational challenge of the Boltzmann collision operator by employing a Fourier Neural Operator (FNO) to learn the evolution operator for both the Boltzmann equation and its BGK simplification. By incorporating conservation-based penalties into a conservative-FNO (C-FNO) loss, the framework enforces mass, momentum, and energy preservation, improving physical fidelity and stability across 1D and 2D kinetic problems. Through extensive 1D BGK and 2D Boltzmann experiments, the approach demonstrates competitive accuracy and notable improvements in conservation properties, with clear guidance on hyperparameter sensitivity (Fourier modes, hidden channels, and layer depth). The work highlights the potential of physics-constrained operator learning for nonlinear integro-differential equations in kinetic theory and beyond, while suggesting future work to embed hard constraints directly into architecture or training dynamics for strict conservation guarantees.
Abstract
The numerical approximation of the Boltzmann collision operator presents significant challenges arising from its high dimensionality, nonlinear structure, and nonlocal integral form. In this work, we propose a Fourier Neural Operator (FNO) based framework to learn the Boltzmann collision operator and its simplified BGK model across different dimensions. The proposed operator learning approach efficiently captures the mapping between the distribution functions in either sequence-to-sequence or point to point manner, without relying on fine grained discretization and large amount of data. To enhance physical consistency, conservation constraints are embedded into the loss functional to enforce improved adherence to the fundamental conservation laws of mass, momentum, and energy compared with the original FNO framework. Several numerical experiments are presented to demonstrate that the modified FNO can efficiently achieve the accurate and physically consistent results, highlighting its potential as a promising framework for physics constrained operator learning in kinetic theory and other nonlinear integro-differential equations.