FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation

• URL: http://arxiv.org/abs/2504.20408v1
• Date: Tue, 29 Apr 2025 04:07:03 GMT
• Title: FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation
• Authors: Jae Yong Lee, Gwang Jae Jung, Byung Chan Lim, Hyung Ju Hwang,
• Abstract summary: We propose a hybrid framework that integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently.<n>FourierSpecNet achieves resolution-invariant learning and supports zero-shot super-resolution, enabling accurate predictions at unseen resolutions without retraining.<n>We evaluate our method on several benchmark cases, including Maxwellian and hard-sphere molecular models, as well as inelastic collision scenarios.
• Score: 10.910310257111414
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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.

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