The Boltzmann equation, a fundamental model in kinetic theory, describes the evolution of particle distribution functions through a nonlinear, high-dimensional collision operator. However, its numerical solution remains computationally demanding, particularly for inelastic collisions and high-dimensional velocity domains. In this work, we propose the Fourier Neural Spectral Network (FourierSpecNet), a hybrid framework that integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently. FourierSpecNet achieves resolution-invariant learning and supports zero-shot super-resolution, enabling accurate predictions at unseen resolutions without retraining. Beyond empirical validation, we establish a consistency result showing that the trained operator converges to the spectral solution as the discretization is refined. We evaluate our method on several benchmark cases, including Maxwellian and hard-sphere molecular models, as well as inelastic collision scenarios. The results demonstrate that FourierSpecNet offers competitive accuracy while significantly reducing computational cost compared to traditional spectral solvers. Our approach provides a robust and scalable alternative for solving the Boltzmann equation across both elastic and inelastic regimes.

该研究解决了玻尔兹曼方程在非弹性碰撞和高维速度域中数值解法计算成本高的问题。提出的傅里叶谱神经网络(FourierSpecNet)将傅里叶谱方法与深度学习结合，在傅里叶空间中高效地近似碰撞算子，能够实现超分辨率预测且不需要重新训练。研究表明，FourierSpecNet 在准确性上具有竞争力，同时显著降低了传统谱解算器的计算成本，为玻尔兹曼方程的求解提供了一种稳健且可扩展的替代方案。

傅里叶谱神经网络：基于傅里叶谱方法的碰撞算子近似以求解玻尔兹曼方程