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

Jae Yong Lee, Gwang Jae Jung, Byung Chan Lim, Hyung Ju Hwang

TL;DR

This work tackles the computational burden of the Boltzmann collision operator by introducing FourierSpecNet, a hybrid neural-operator that learns in Fourier space to approximate $Q(f,f)$ with resolution-invariant capabilities. By constraining neural parameters to the lowest $N_{trun}^d$ Fourier modes and leveraging a decoupled fast spectral decomposition, the method achieves zero-shot super-resolution and GPU-accelerated inference while preserving mass, momentum, and energy. A theoretical consistency bound shows the learned operator converges to the spectral solution as the discretization is refined, bridging deep learning with classical spectral solvers. Empirical results across Maxwellian, hard-sphere, and inelastic regimes, including 3D velocity space, demonstrate competitive accuracy with substantial computational gains, highlighting the method's potential for scalable kinetic simulations in diverse regimes.

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.

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

TL;DR

This work tackles the computational burden of the Boltzmann collision operator by introducing FourierSpecNet, a hybrid neural-operator that learns in Fourier space to approximate with resolution-invariant capabilities. By constraining neural parameters to the lowest Fourier modes and leveraging a decoupled fast spectral decomposition, the method achieves zero-shot super-resolution and GPU-accelerated inference while preserving mass, momentum, and energy. A theoretical consistency bound shows the learned operator converges to the spectral solution as the discretization is refined, bridging deep learning with classical spectral solvers. Empirical results across Maxwellian, hard-sphere, and inelastic regimes, including 3D velocity space, demonstrate competitive accuracy with substantial computational gains, highlighting the method's potential for scalable kinetic simulations in diverse regimes.

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.
Paper Structure (18 sections, 6 theorems, 67 equations, 11 figures, 1 table)

This paper contains 18 sections, 6 theorems, 67 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Boltzmann equation
  4. Fundamental properties of physical quantities and the equilibrium state
  5. Related works
  6. Outline of the paper
  7. Fourier Spectral Method
  8. Proposed model: Fourier neural spectral network (FourierSpecNet)
  9. Training of FourierSpecNet
  10. FourierSpecNet with zero-shot super-resolution capabilities
  11. Properties of the Fourier neural spectral network
  12. Simulation Results
  13. Maxwellian molecules case
  14. Hard sphere molecules case
  15. Inelastic Boltzmann case
  16. ...and 3 more sections

Key Result

Proposition 3.1

Suppose $r \geq 0$ and the collision kernel satisfies $\hat{G}(l, m) \rightarrow 0$ as $\| l \|_\infty \rightarrow \infty$ or $\| m \|_\infty \rightarrow \infty$. Then for any $\varepsilon > 0$, there exist such that for any $f \in L^\infty(D_T)$ and any $N \in \mathbb{N}$, the following error bound holds: where $C > 0$ is a constant depending on $T$ and $\| f \|_{L^2(D_T)}$. Furthermore, if $f

Figures (11)

  • Figure 1: Overview of the proposed Fourier neural spectral network to approximate the collision operator of the Boltzmann equation. Note that the size of learnable network parameters $\{\alpha^{nn}_t\}_{t=1}^M$, $\{\beta^{nn}_t\}_{t=1}^M$, and $\{\gamma^{nn}_t\}_{t=1}^M$ are independent of the size of input $f(\boldsymbol{v})$.
  • Figure 2: Illustration of the proposed Fourier Neural Spectral Network as a super-resolution approximator for the Boltzmann collision operator. The model ($\alpha, \beta, \gamma$) is trained on low-resolution input-output pairs ($f(v_1, v_2)$ and $Q(v_1, v_2)$) defined on a $16 \times 16$ grid. During inference, the model generalizes to high-resolution inputs ($128 \times 128$ grid) and accurately predicts the corresponding high-resolution outputs, demonstrating its resolution-invariance and super-resolution capabilities.
  • Figure 3: Time evolution of the BKW solution for Maxwellian molecules. The top row shows the pointwise predicted values of $f^{nn}(t, \boldsymbol{v})$ (FourierSpecNet) over time ($t = 0$ to $t = 5$). The bottom row illustrates detailed 2D slices of the velocity space at specific time instances, comparing FourierSpecNet predictions with the exact analytical solution (dotted lines) and numerical results (solid lines).
  • Figure 4: Physical quantities of the BKW solution ($f^{nn}$) as a function of time ($t = 0$ to $t = 5$). The predicted values (green lines) for mass, momentum ($v_1$ and $v_2$), kinetic energy, and entropy are plotted.
  • Figure 5: Evaluation of the resolution-invariance and super-resolution capability of FourierSpecNet. The left panel shows the predicted collision operator $Q(f, f)$ for various grid resolutions ($N = 16, 32, 64, 128$), demonstrating the model's ability to generalize across different resolutions. The right panel presents the relative $L^2$-error of FourierSpecNet (denoted as 'Ours') and the spectral method over time ($t = 0$ to $t = 5$), highlighting FourierSpecNet's consistent accuracy across all resolutions.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Proposition 3.1: Consistency of FourierSpecNet
  • Remark
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop:consistency_gjjung']}
  • Lemma A.1: Decay of the kernel modes
  • Lemma A.2
  • Lemma A.3
  • ...and 2 more