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Domain decomposition for data-driven reduced modeling of large-scale systems

Ionut-Gabriel Farcas, Rayomand P. Gundevia, Ramakanth Munipalli, Karen E. Willcox

TL;DR

The results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure and up to 5% for other key quantities, such as temperature, and fuel, and oxidizer mass fractions.

Abstract

This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training datasets. In these settings, standard, single-domain approaches may be too inaccurate or may overfit and hence generalize poorly. Moreover, processing large-scale datasets typically requires significant memory and computing resources which can render single-domain approaches computationally prohibitive. To address these challenges, we introduce a domain decomposition formulation into the construction of a data-driven reduced model. In doing so, the basis functions used in the reduced model approximation become localized in space, which can increase the accuracy of the domain-decomposed approximation of the complex dynamics. The decomposition furthermore reduces the memory and computing requirements to process the underlying large-scale training dataset. We demonstrate the effectiveness and scalability of our approach in a large-scale three-dimensional unsteady rotating detonation rocket engine simulation scenario with over $75$ million degrees of freedom and a sparse training dataset. Our results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure by up to $13 \%$ and up to $5\%$ for other key quantities, such as temperature, and fuel and oxidizer mass fractions. Lastly, our approach decreases the memory requirements for processing by almost a factor of four, which in turn reduces the computing requirements as well.

Domain decomposition for data-driven reduced modeling of large-scale systems

TL;DR

The results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure and up to 5% for other key quantities, such as temperature, and fuel, and oxidizer mass fractions.

Abstract

This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training datasets. In these settings, standard, single-domain approaches may be too inaccurate or may overfit and hence generalize poorly. Moreover, processing large-scale datasets typically requires significant memory and computing resources which can render single-domain approaches computationally prohibitive. To address these challenges, we introduce a domain decomposition formulation into the construction of a data-driven reduced model. In doing so, the basis functions used in the reduced model approximation become localized in space, which can increase the accuracy of the domain-decomposed approximation of the complex dynamics. The decomposition furthermore reduces the memory and computing requirements to process the underlying large-scale training dataset. We demonstrate the effectiveness and scalability of our approach in a large-scale three-dimensional unsteady rotating detonation rocket engine simulation scenario with over million degrees of freedom and a sparse training dataset. Our results show that compared to the single-domain approach, the domain-decomposed version reduces both the training and prediction errors for pressure by up to and up to for other key quantities, such as temperature, and fuel and oxidizer mass fractions. Lastly, our approach decreases the memory requirements for processing by almost a factor of four, which in turn reduces the computing requirements as well.
Paper Structure (15 sections, 16 equations, 15 figures)

This paper contains 15 sections, 16 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Single-domain physics-based data-driven reduced modeling
  3. Setup for large-scale high-fidelity simulations
  4. Learning physics-based data-driven reduced models via Operator Inference
  5. Domain-decomposed physics-based data-driven reduced modeling
  6. Domain-decomposed formulation of the governing equations
  7. Domain-decomposed Operator Inference
  8. General considerations
  9. Learning the domain-decomposed reduced operators via Operator Inference
  10. Obtaining the full-domain solution using the domain-decomposed reduced solutions
  11. Numerical demonstration in a large-scale rotating detonation rocket engine simulation
  12. Problem setup for high-fidelity large-eddy simulations
  13. Setup for data-driven reduced modeling
  14. Results using domain-decomposed Operator Inference
  15. Conclusions

Figures (15)

  • Figure 1: Two example domains decomposed into three overlapping subdomains. The overlapping regions are shown in red.
  • Figure 2: Visual illustration of the process of determining the full-domain solution using domain-decomposed solutions using an example with three overlapping subdomains.
  • Figure 3: Pressure field example (top), and the $x$-$y$ and $y$-$z$ extents of the computational domain (bottom). For more details, we refer the reader to Ref. Be23.
  • Figure 4: Combustion chamber decomposition into four overlapping sector with overlapping region between two sectors of $\pi/9$ radians.
  • Figure 5: POD singular values (top left) and retained energy (top right) for SD OpInf. POD singular values (bottom left) and total energy (bottom right) for both DD-$4$ and SD OpInf.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5