Reducing Training Complexity in Empirical Quadrature-Based Model Reduction via Structured Compression

TL;DR

This work tackles the high offline cost of training complexity-reduction surrogates for nonlinear reduced-order models by introducing a preprocessing step that compresses training data along the snapshot dimension. By factorizing the data into structured low-rank forms and solving a constrained CPCA problem, the method reduces the training problem’s effective size from $K\cdot N_r$ to a much smaller $K_t$, while preserving accuracy via rigorous a posteriori and a priori bounds. The approach enables scalable application of empirical quadrature and cell-based empirical cubature in large-scale FE-based problems, demonstrated through substantial offline-time and memory savings with only minor loss in fidelity. The numerical experiments on a 3D reaction-diffusion system and a gas-transport network illustrate meaningful speedups and reliable error control, making nonlinear hyper-reduction more practical for large-scale simulations.

Abstract

Model order reduction seeks to approximate large-scale dynamical systems by lower-dimensional reduced models. For linear systems, a small reduced dimension directly translates into low computational cost, ensuring online efficiency. This property does not generally hold for nonlinear systems, where an additional approximation of nonlinear terms -- known as complexity reduction -- is required. To achieve online efficiency, empirical quadrature and cell-based empirical cubature are among the most effective complexity reduction techniques. However, existing offline training algorithms can be prohibitively expensive because they operate on raw snapshot data of all nonlinear integrands associated with the reduced model. In this paper, we introduce a preprocessing approach based on a specific structured compression of the training data. Its key feature is that it scales only with the number of collected snapshots, rather than additionally with the reduced model dimension. Overall, this yields roughly an order-of-magnitude reduction in offline computational cost and memory requirements, thereby enabling the application of the complexity reduction methods to larger-scale problems. Accuracy is preserved, as indicated by our error analysis and demonstrated through numerical examples.

Figures (5)

• Figure 1: Visualization of the spatial domains for both numerical benchmarks. Left: reaction-diffusion equation domain plus $M_c=50$ trained empirical quadrature points; right: gas network topology from art:bls-snapBasedMorComp with boundary nodes highlighted in red.
• Figure 2: Reaction-diffusion equation with ${C} =0.75$: FOM solution visualized by slices of width 0.2 at different time points. (FOM with $163\,840$ cells.)
• Figure 3: Reaction-diffusion equation. Training performance for varying $M_c$, with and without compression. Left: Number of equations in offline training (independent of FOM dimension). Right: Relative space-time $L_2$-errors of CROM for the scenario with ${C} =0.75$ and FOM with $56{\,}025$ cells.
• Figure 4: Reaction-diffusion equation. Offline training times with and without compression, the varying number of cells in the FOM and $M_c$.
• Figure 5: Gas network. Comparison of training performance for varying $M_c$, with and without compression (using ${{K}_{\rm{t}}} = M_c+10$). Left: offline training times; right: training residuals, the difference introduced by compression, and the corresponding a posteriori bound from \ref{['the:CompressedOptFun']} (scaled by $\|{\mathbf g}\|$, where ${\mathbf g}$ denotes the right-hand side of the training problem).

Theorems & Definitions (13)

• remark 1.1
• remark 2.1
• remark 2.2
• remark 4.1
• Lemma 4.2
• Proof 1
• Corollary 4.3
• Corollary 4.4
• remark 4.5
• remark 5.1