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Adaptive and hybrid reduced order models to mitigate Kolmogorov barrier in a multiscale kinetic transport equation

Tianyu Jin, Zhichao Peng, Yang Xiang

TL;DR

This work tackles efficient reduced order modeling for the time-dependent radiative transport equation with diffusion limits, where transport-dominant regimes cause slow decay of the Kolmogorov $n$-width and degrade linear ROM performance. It develops a piecewise linear ROM built on a goal-oriented adaptive time partitioning strategy, augmented with a coarsening mechanism and an equilibrium-detection step to maintain ultra-low-dimensional spaces as systems approach steady state. To handle intervals where linear reduction is insufficient, the authors introduce a hybrid ROM that selectively applies an autoencoder-based nonlinear ROM, dramatically reducing offline training cost by focusing training on the most challenging time windows. Comprehensive numerical experiments across two-, three-material, and 2D pin-cell problems demonstrate that the adaptive partitioning and the autoencoder-assisted hybrid ROM achieve higher accuracy and significantly lower online computation times compared with traditional POD-based approaches, providing a first systematic treatment of the Kolmogorov barrier for multiscale kinetic transport problems with mixed transport- and diffusion-dominated behavior.

Abstract

In this work, we develop reduced order models (ROMs) to predict solutions to a multiscale kinetic transport equation with a diffusion limit under the parametric setting. When the underlying scattering effect is not sufficiently strong, the system governed by this equation exhibits transport-dominated behavior. Suffering from the Kolmogorov barrier for transport-dominant problems, classical linear ROMs may become inefficient in this regime. To address this issue, we first develop a piecewise linear ROM by introducing a novel goal-oriented adaptive time partitioning strategy. To avoid local over-refinement or under-refinement, we propose an adaptive coarsening and refinement strategy that remains robust with various initial empirical partitions. Additionally, for problems where a local linear approximation is not sufficiently efficient, we further develop a hybrid ROM, which combines autoencoder-based nonlinear ROMs and piecewise linear ROMs. Compared to previous autoencoder-based ROMs, this hybridized method reduces the offline autoencoder's training cost by only applying it to time intervals that are adaptively identified as the most challenging. Numerical experiments demonstrate that our proposed approaches successfully predict full-order solutions at unseen parameter values with both efficiency and accuracy. To the best of our knowledge, this is the first attempt to address the Kolmogorov barrier for multiscale kinetic transport problems with the coexistence of both transport- and diffusion-dominant behaviors.

Adaptive and hybrid reduced order models to mitigate Kolmogorov barrier in a multiscale kinetic transport equation

TL;DR

This work tackles efficient reduced order modeling for the time-dependent radiative transport equation with diffusion limits, where transport-dominant regimes cause slow decay of the Kolmogorov -width and degrade linear ROM performance. It develops a piecewise linear ROM built on a goal-oriented adaptive time partitioning strategy, augmented with a coarsening mechanism and an equilibrium-detection step to maintain ultra-low-dimensional spaces as systems approach steady state. To handle intervals where linear reduction is insufficient, the authors introduce a hybrid ROM that selectively applies an autoencoder-based nonlinear ROM, dramatically reducing offline training cost by focusing training on the most challenging time windows. Comprehensive numerical experiments across two-, three-material, and 2D pin-cell problems demonstrate that the adaptive partitioning and the autoencoder-assisted hybrid ROM achieve higher accuracy and significantly lower online computation times compared with traditional POD-based approaches, providing a first systematic treatment of the Kolmogorov barrier for multiscale kinetic transport problems with mixed transport- and diffusion-dominated behavior.

Abstract

In this work, we develop reduced order models (ROMs) to predict solutions to a multiscale kinetic transport equation with a diffusion limit under the parametric setting. When the underlying scattering effect is not sufficiently strong, the system governed by this equation exhibits transport-dominated behavior. Suffering from the Kolmogorov barrier for transport-dominant problems, classical linear ROMs may become inefficient in this regime. To address this issue, we first develop a piecewise linear ROM by introducing a novel goal-oriented adaptive time partitioning strategy. To avoid local over-refinement or under-refinement, we propose an adaptive coarsening and refinement strategy that remains robust with various initial empirical partitions. Additionally, for problems where a local linear approximation is not sufficiently efficient, we further develop a hybrid ROM, which combines autoencoder-based nonlinear ROMs and piecewise linear ROMs. Compared to previous autoencoder-based ROMs, this hybridized method reduces the offline autoencoder's training cost by only applying it to time intervals that are adaptively identified as the most challenging. Numerical experiments demonstrate that our proposed approaches successfully predict full-order solutions at unseen parameter values with both efficiency and accuracy. To the best of our knowledge, this is the first attempt to address the Kolmogorov barrier for multiscale kinetic transport problems with the coexistence of both transport- and diffusion-dominant behaviors.
Paper Structure (17 sections, 1 theorem, 24 equations, 12 figures, 12 tables)

This paper contains 17 sections, 1 theorem, 24 equations, 12 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Problem setting and preliminaries
  3. Reduced order model and proper orthogonal decomposition
  4. Proper orthogonal decomposition
  5. A non-intrusive online stage
  6. Kolmogorov barrier for ROM construction
  7. Offline time partitioning to mitigate the Kolmogorov barrier
  8. Principal interval decomposition (uniform partitioning)
  9. Goal-oriented adaptive time partitioning
  10. Hybrid ROM with time partitioning
  11. Numerical experiments
  12. Example 1: two-material problem
  13. Example 2: three-material problem
  14. Example 3: pin-cell problem
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 3.1

(Reconstruction error.) Given the snapshot matrix $S\in\mathbb{R}^{n_h\times n_s}$ and partition its columns into $k$ parts, then we obtain $k$ submatrices $S_j\in\mathbb{R}^{n_h\times n_j}, j=1,2,\dots,k$ and $\sum_{j=1}^k n_j = n_s$, i.e. $S = (S_1|\dots|S_k)$. Suppose for any $j = 1,2,\dots, k$, then the relative reconstruction error for the whole snapshot matrix satisfies

Figures (12)

  • Figure 1: Complexity of classical POD and uniform time partition into $k$ intervals in Example 1 when varying $k$, $\varepsilon_{\text{POD}}=10^{-4}$. (a) Average number of bases regarding online time; (b) Total number of bases regarding memory cost.
  • Figure 2: Architecture of convolutional autoencoder: the purple blocks represent the convolutional and transposed convolutional layers, while the green blocks represent the activation function.
  • Figure 3: Reference solutions for Example 1 corresponding to $\mu=5$.
  • Figure 4: Number of basis for various adaptive time partitioning strategies for Example 1.
  • Figure 5: How time partition evolves in Example 1 with different uniform initial partitions. Left: $4$ initial intervals. Right: $8$ initial intervals.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • proof