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

arXiv: https://arxiv.org/abs/2504.20408 · HTML: https://mlsanity.com/paper/2504.20408/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 45 references that are in the map.
- First posted on arXiv 29 Apr 2025.

## Builds on (most cited first, 7 of 7)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Finite Element Operator Network for Solving Elliptic-type parametric PDEs](https://mlsanity.com/paper/2308.04690/) (2023)
- [Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach](https://mlsanity.com/paper/1911.09843/) (2019)
- [opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation](https://mlsanity.com/paper/2207.01765/) (2022)
- [Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation](https://mlsanity.com/paper/2403.06342/) (2024)
- [The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach](https://mlsanity.com/paper/2009.13280/) (2020)

## Built on by (most cited first, 1 of 1)

- [Learning collision operators from plasma phase space data using differentiable simulators](https://mlsanity.com/paper/2601.10885/) (2026)

## Abstract (arXiv)

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.