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Randomized Proper Orthogonal Decomposition for data-driven reduced order modeling of a two-layer quasi-geostrophic ocean model

Lander Besabe, Michele Girfoglio, Annalisa Quaini, Gianluigi Rozza

TL;DR

This work tackles the high computational cost of simulating the two-layer quasi-geostrophic equations (2QGE) across parameter spaces by developing a data-driven reduced-order model (ROM) that combines randomized POD (rPOD) with long short-term memory (LSTM) networks. A nonlinear Helmholtz filtering stabilization (2QG-NL-$\alpha$) enables cheaper snapshot generation on meshes larger than the Munk scale, while rPOD accelerates basis construction and LSTM forecasts the modal coefficients for online predictions. The approach achieves large speedups (offline ~700×, online ~4.7×10^5×) and retains accuracy for time-averaged fields across up to three varying parameters; performance degrades with a fourth parameter, highlighting the need for finer sampling or ROM closures. The methodology offers a practical path to rapid parametric studies and uncertainty quantification for geophysical flow models, with potential extensions to adaptive sampling and closure models to further enhance accuracy in high-dimensional parameter spaces.

Abstract

The two-layer quasi-geostrophic equations (2QGE) serve as a simplified model for simulating wind-driven, stratified ocean flows. However, their numerical simulation remains computationally expensive due to the need for high-resolution meshes to capture a wide range of turbulent scales. This becomes especially problematic when several simulations need to be run because of, e.g., uncertainty in the parameter settings. To address this challenge, we propose a data-driven reduced order model (ROM) for the 2QGE that leverages randomized proper orthogonal decomposition (rPOD) and long short-term memory (LSTM) networks. To efficiently generate the snapshot data required for model construction, we apply a nonlinear filtering stabilization technique that allows for the use of larger mesh sizes compared to a direct numerical simulations (DNS). Thanks to the use of rPOD to extract the dominant modes from the snapshot matrices, we achieve up to 700 times speedup over the use of deterministic POD. LSTM networks are trained with the modal coefficients associated with the snapshots to enable the prediction of the time- and parameter-dependent modal coefficients during the online phase, which is hundreds of thousands of time faster than a DNS. We assess the accuracy and efficiency of our rPOD-LSTM ROM through an extension of a well-known benchmark called double-gyre wind forcing test. The dimension of the parameter space in this test is increased from two to four.

Randomized Proper Orthogonal Decomposition for data-driven reduced order modeling of a two-layer quasi-geostrophic ocean model

TL;DR

This work tackles the high computational cost of simulating the two-layer quasi-geostrophic equations (2QGE) across parameter spaces by developing a data-driven reduced-order model (ROM) that combines randomized POD (rPOD) with long short-term memory (LSTM) networks. A nonlinear Helmholtz filtering stabilization (2QG-NL-) enables cheaper snapshot generation on meshes larger than the Munk scale, while rPOD accelerates basis construction and LSTM forecasts the modal coefficients for online predictions. The approach achieves large speedups (offline ~700×, online ~4.7×10^5×) and retains accuracy for time-averaged fields across up to three varying parameters; performance degrades with a fourth parameter, highlighting the need for finer sampling or ROM closures. The methodology offers a practical path to rapid parametric studies and uncertainty quantification for geophysical flow models, with potential extensions to adaptive sampling and closure models to further enhance accuracy in high-dimensional parameter spaces.

Abstract

The two-layer quasi-geostrophic equations (2QGE) serve as a simplified model for simulating wind-driven, stratified ocean flows. However, their numerical simulation remains computationally expensive due to the need for high-resolution meshes to capture a wide range of turbulent scales. This becomes especially problematic when several simulations need to be run because of, e.g., uncertainty in the parameter settings. To address this challenge, we propose a data-driven reduced order model (ROM) for the 2QGE that leverages randomized proper orthogonal decomposition (rPOD) and long short-term memory (LSTM) networks. To efficiently generate the snapshot data required for model construction, we apply a nonlinear filtering stabilization technique that allows for the use of larger mesh sizes compared to a direct numerical simulations (DNS). Thanks to the use of rPOD to extract the dominant modes from the snapshot matrices, we achieve up to 700 times speedup over the use of deterministic POD. LSTM networks are trained with the modal coefficients associated with the snapshots to enable the prediction of the time- and parameter-dependent modal coefficients during the online phase, which is hundreds of thousands of time faster than a DNS. We assess the accuracy and efficiency of our rPOD-LSTM ROM through an extension of a well-known benchmark called double-gyre wind forcing test. The dimension of the parameter space in this test is increased from two to four.

Paper Structure

This paper contains 13 sections, 27 equations, 11 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Full Order Method
  4. Reduced Order Modeling
  5. Proper Orthogonal Decomposition
  6. Randomized Proper Orthogonal Decomposition
  7. Long short-term memory network
  8. Numerical Results
  9. ROM for two varying parameters
  10. ROM for three varying parameters
  11. ROM for four varying parameters
  12. Computational time reduction
  13. Conclusions

Figures (11)

  • Figure 1: Computational time required to apply POD to a snapshot matrix associated to each variable for dimension of the physical parameter space $d$ ranging from 0 (i.e., time is the only parameter) to 3 (i.e., $\delta$, $\sigma$, and $Fr$, in addition to time).
  • Figure 2: Decay of singular values from rPOD with target rank $N_\Phi^r = 10$ and $p=0,5,10,20,50,75$ for variables $q_1$ (top left), $q_2$ (top right),$\psi_1$ (bottom left) and $\psi_2$ (bottom right).
  • Figure 3: First (left panel) and tenth (right panel) modes of $q_1$ (first row) and $q_2$ (second row) computed using POD (first column in each panel) and rPOD with $p=0$ (second column in each panel) and oversampling of $p=75$ (third column in each panel). The fourth column in each panel shows the absolute difference between modes from POD and rPOD with $p=75$.
  • Figure 4: First (left panel) and tenth (right panel) modes of $\psi_1$ (first row) and $\psi_2$ (second row) computed using POD (first column in each panel) and rPOD with $p=0$ (second column in each panel) and oversampling of $p=75$ (third column in each panel). The fourth column in each panel shows the absolute difference between modes from POD and rPOD with $p=75$.
  • Figure 5: Computational time needed by the POD algorithm and the rPOD algorithm (left) and the corresponding speedup factor (right) as $p$ varies.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Remark