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Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations

Jae Yong Lee, Steffen Schotthöfer, Tianbai Xiao, Sebastian Krumscheid, Martin Frank

TL;DR

The paper addresses learning the collision operator $Q(f)$ in the linear Boltzmann equation while preserving physical structure, notably collision invariants and entropy. It adopts DeepONet and introduces an input-dependent bias and basis-augmentation plus quadrature-based constraints to enforce conservation laws during training. Through a toy isotropic kernel and four kernel/settings (isotropic linear, Henyey–Greenstein, nonlinear, BGK), the approach demonstrates accurate $Q(f)$ approximation alongside preserved mass, momentum, and energy and improved entropy behavior. This structure-preserving operator learning enables stable, physically consistent surrogate modeling of kinetic equations with potential broad impact on high-dimensional particle transport simulations.

Abstract

This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation.

Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations

TL;DR

The paper addresses learning the collision operator in the linear Boltzmann equation while preserving physical structure, notably collision invariants and entropy. It adopts DeepONet and introduces an input-dependent bias and basis-augmentation plus quadrature-based constraints to enforce conservation laws during training. Through a toy isotropic kernel and four kernel/settings (isotropic linear, Henyey–Greenstein, nonlinear, BGK), the approach demonstrates accurate approximation alongside preserved mass, momentum, and energy and improved entropy behavior. This structure-preserving operator learning enables stable, physically consistent surrogate modeling of kinetic equations with potential broad impact on high-dimensional particle transport simulations.

Abstract

This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation.
Paper Structure (19 sections, 2 theorems, 25 equations, 4 figures, 1 table)

This paper contains 19 sections, 2 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Boltzmann Equation [Tianbai and Steffen]
  3. Structure of the Collision Operator
  4. DeepOnet Theory [JaeYong]
  5. Toy example
  6. Setting
  7. Ideas
  8. Jaeyong Idea
  9. Sebastian Idea - Toy example (basic/vanilla version).
  10. Tianbai Idea - The idea of preserving the conservation property
  11. Experiment results
  12. Future works
  13. Structure Preservation
  14. Numerical Results
  15. Isotropic linear Collision
  16. ...and 4 more sections

Key Result

theorem 1

The linear collision operator of the Boltzmann equation $Q$ with positive collision kernel $k>0$ and the property $k(\mathbf{v},\mathbf{v}_*)=k(\mathbf{v}_*,v)$ fulfills for a convex entropy density $\eta$ and a kinetic density $f$ the dissipation property where $\text{Img}(f)$ denotes the image of $f$ and $\text{Dom}( Q )$ denotes the domain of $Q$.

Figures (4)

  • Figure 1: The structure of vanila DeepONet \ref{['eq_vanila_deeponet']}. This original model is unstacked DeepONet with an additional bias proposed in lu2021learning.
  • Figure 2: The structure of modified DeepONet \ref{['eq_modify_deeponet']}.
  • Figure 3: Basis functions.
  • Figure 4: 3 test samples.

Theorems & Definitions (3)

  • definition 1: Collision Invariant
  • theorem 1: H-Theorem for the Linear Collision Operator
  • theorem 2: Gallilean Invariance Junk2002MaximumEM