Improved Greedy Identification of Latent Dynamics with Application to Fluid Flows

TL;DR

This paper presents I-GILD, an improved method for identifying quadratic reduced-order models from data, which enhances convergence and accuracy over previous approaches, demonstrated on fluid flow problems.

Contribution

The paper introduces I-GILD, a refined regression technique that reformulates the model learning problem, leading to faster convergence and better accuracy in data-driven reduced-order modeling.

Findings

I-GILD converges faster than GILD with fewer iterations.

The method effectively models complex fluid flows from experimental data.

Error bounds provide insights into model prediction accuracy over time.

Abstract

Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that subsequently allows control, optimization, and analysis with multi-query algorithms. We introduce an enhanced regression technique tailored to uncover quadratic parametric reduced-order dynamical systems from data. Our method, termed Improved Greedy Identification of Latent Dynamics (I-GILD), refines the learning phase of the original GILD approach. This refinement is achieved by reorganizing the quadratic model coefficients, allowing the minimum-residual problem to be reformulated using the Frobenius norm. Consequently, the optimality conditions lead to a generalized Sylvester equation, which is efficiently solved using the conjugate gradient method. Analysis of the convergence shows that I-GILD achieves superior convergence for quadratic model coefficients compared to GILD's steepest gradient descent, reducing both computational complexity and iteration count. Additionally, we derive an error bound for the model predictions, offering insights into error growth in time and ensuring controlled accuracy as long as the magnitudes of initial error is small and learning residuals are well minimized. The efficacy of I-GILD is demonstrated through its application to numerical and experimental tests, specifically the flow past Ahmed body with a variable rear slant angle, and the lid-driven cylindrical cavity problem with variable Reynolds numbers, utilizing particle-image velocimetry (PIV) data. These tests confirm I-GILD's ability to treat real-world dynamical system challenges and produce effective reduced-order models.