Structure-preserving neural networks for the regularized entropy-based closure of the Boltzmann moment system

TL;DR

The paper tackles the memory and computation bottlenecks of solving high-dimensional kinetic (Boltzmann) moment systems by formulating a structure-preserving neural network surrogate for the regularized entropy-based closure. It introduces a partially regularized entropy framework to preserve scaling while regularizing the problematic components, and uses input-convex neural networks to approximate the reduced entropy function and its gradient, ensuring entropy dissipation and hyperbolicity in the closed system. An explicit error analysis partitions neural-network approximation error, regularization error, and scaling error, with extensive 2D numerical tests on line-source and hohlraum benchmarks demonstrating reduced memory footprint and competitive accuracy relative to $P_N$ and $S_N$ methods, integrated within the KiT-RT solver. The work shows that neural surrogates can accelerate higher-order closures without sacrificing core physical structure, offering a practical route for memory-efficient radiation transport simulations.

Abstract

The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the context of regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy closure as a two-stage approximation to the original entropy closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy.

Figures (15)

• Figure 1: The sets of sampled moments ${\overline{\mathbf{u}}_{\#}}$ and the regularized entropy functions $\hat{h}^{\gamma}({\overline{\mathbf{u}}_{\#}})$, see Theorem \ref{['thm_reg_rescale']}, for different regularization parameters $\gamma$. The moments are computed from multipliers ${{\boldsymbol{\beta}}}$, i.e., ${\overline{\mathbf{u}}_{\#}}=\boldsymbol{\psi}^\gamma({{\boldsymbol{\beta}}})$, where $\boldsymbol{\psi}^\gamma$ is defined in \ref{['eq_u_from_alpha']} and ${{\boldsymbol{\beta}}}$ is sampled from the set $\left\lbrace{{\boldsymbol{\beta}}}\in\mathbb{R}^{n}:\left\lVert\boldsymbol{\beta}\right\rVert<M \right\rbrace$.
• Figure 2: Scatter plots for sampled values of ${\boldsymbol{\beta}}$ (bottom row) and the corresponding values of ${\overline{\mathbf{u}}_{\#}}=\boldsymbol{\psi}^\gamma({{\boldsymbol{\beta}}})$ (top row), where $\boldsymbol{\psi}^\gamma$ is defined in \ref{['eq_u_from_alpha']}. The value of $\hat{h}^\gamma$ is represented in each plot by a heatmap. Less regularization leads to steeper slopes of $\hat{h}^\gamma$ and thus higher sampling densities in regions where $\left\lVert\overline{{\mathbf{u}}}\right\rVert$ is large. The sampling strategy is described in Algorithm \ref{['alg_sampling_entropy']} using the set $B_{M=40,\tau=0.01}^\gamma$, defined in \ref{['eq:BMtau']}, for the rejection criteria.
• Figure 3: Comparison of ICNN and ResNet-based test errors for M$_3$ (a)-(f) and M$_4$ (g)-(l) closures, with different regularization levels $\gamma$. The reported error $e_{\hat{h}^\gamma}$, $e_{{\boldsymbol{\beta}}^\gamma_{{\overline{\mathbf{u}}_{\#}}}}$ and ${e}_{{\overline{\mathbf{u}}_{\#}}}$ are defined inEq. \ref{['eq_test_error_h']}, \ref{['eq_test_error_beta']}, and \ref{['eq_test_error_u']}, respectively. The lowest test error up to the current epoch is plotted for each choice of $\gamma$. The test errors of the ICNN model are reduced heavily by increasing $\gamma$, whereas ResNet test errors reduce only slightly. Results for the M$_2$ closure are comparable.
• Figure 4: Line source test case. (a) ICNN-based M$_2$ simulation with $\gamma=1e-3$ of the line source test case. The test case dynamic at $t_f$ is captured well by the ICNN-simulation. At the wavefront, slight asymmetries are observed since rotational invariance is not enforced by the neural network-based simulation. (b) Rotatationally invariant ICNN-based simulation with $\gamma=1e-3$ of the line source test case. (c) Reference solution computed with a Newton optimizer.
• Figure 5: Setup and reference solution for the hohlraum test case.

Theorems & Definitions (32)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• proof
• Lemma 1
• proof
• Definition 3
• Lemma 2
• proof