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.

翻译：玻尔兹曼碰撞算子的数值逼近面临着高维性、非线性结构及非局部积分形式带来的重大挑战。本文提出一种基于傅里叶神经算子（FNO）的框架，用于学习不同维度下的玻尔兹曼碰撞算子及其简化BGK模型。所提出的算子学习方法以序列到序列或点到点的方式，有效捕捉分布函数间的映射关系，无需依赖精细网格离散化和海量数据。为增强物理一致性，我们在损失函数中嵌入守恒约束，相较于原始FNO框架，该方法能更好地遵循质量、动量和能量的基本守恒定律。通过若干数值实验证明，改进后的FNO能够高效获得精确且物理一致的结果，凸显了其作为动理学理论及其他非线性积分微分方程中物理约束算子学习框架的潜力。