[Paper Review] Data-driven modal decomposition methods as feature detection techniques for flow problems: a critical assessment

Beka Begiashvili, Nourelhouda Groun+3 more|arXiv (Cornell University)|Dec 23, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents a comprehensive, fair comparison of eight data-driven modal decomposition techniques—POD, DMD, FFT, SPOD, HODMD, mPOD, mrDMD, and Resolvent Analysis—for feature detection in fluid flows. It evaluates their performance across laminar, turbulent, and transient flow cases, demonstrating that reconstruction accuracy and computational efficiency vary significantly by method and flow regime, with HODMD and SPOD showing superior robustness in complex scenarios.

ABSTRACT

Modal decomposition techniques are showing a fast growth in popularity for their good properties as data-driven tools. There are several modal decomposition techniques, yet Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are considered the most demanded methods, especially in the field of fluid dynamics. Following their magnificent performance on various applications in several fields, numerous extensions of these techniques have been developed. In this work we present an ambitious review comparing eight different modal decomposition techniques, including most established methods: POD, DMD and Fast Fourier Trasform (FFT), extensions of these classical methods: based on time embedding systems, Spectral POD (SPOD) and Higher Order DMD (HODMD), based on scales separation, multi-scale POD (mPOD), multi-resolution DMD (mrDMD), and based on the properties of the resolvent operator, the data-driven Resolvent Analysis (RA). The performance of all these techniques will be evaluated on three different testcases: the laminar wake around cylinder, a turbulent jet flow, and the three dimensional wake around cylinder in transient regime. First, we show a comparison between the performance of the eight modal decomposition techniques when the datasets are shortened. Next, all the results obtained will be explained in details, showing both the conveniences and inconveniences of all the methods under investigation depending on the type of application and the final goal (reconstruction or identification of the flow physics). In this contribution we aim on giving a -- as fair as possible -- comparison of all the techniques investigated. To the authors knowledge, this is the first time a review paper gathering all this techniques have been produced, clarifying to the community what is the best technique to use for each application.

Motivation & Objective

• To provide a fair, systematic comparison of eight data-driven modal decomposition techniques for fluid flow analysis.
• To evaluate the effectiveness of each method in reconstructing flow fields and identifying underlying physics across diverse flow regimes.
• To clarify the strengths and limitations of each technique based on reconstruction error, computational cost, and physical interpretability.
• To guide practitioners in selecting the optimal method for specific applications, such as flow reconstruction or feature detection.
• To establish a unified framework for comparing modal decomposition techniques, addressing the lack of comprehensive reviews in the literature.

Proposed method

• The study evaluates eight methods: POD, DMD, FFT, SPOD, HODMD, mPOD, mrDMD, and Resolvent Analysis, using three benchmark flow cases: laminar wake, turbulent jet, and 3D transient cylinder wake.
• Reconstruction accuracy is quantified using the Frobenius norm of the difference between the original and reconstructed flow fields, with error minimized via least-squares amplitude estimation.
• Amplitude computation for DMD and HODMD is performed using both direct projection (Eq. 37) and full matrix-based optimization (Eq. 39), with the latter preferred due to superior accuracy.
• For HODMD, amplitudes are computed by solving an over-determined system via pseudo-inverse using SVD, with alternative formulations exploiting matrix structure to reduce memory load.
• A growth-rate-based criterion (Eq. 46) is applied to rank DMD modes by their physical relevance, accounting for both amplitude and temporal dynamics.
• The performance of each method is assessed across three test cases with varying data lengths, focusing on reconstruction error and computational time.

Experimental results

Research questions

• RQ1 Which modal decomposition method achieves the highest reconstruction accuracy across different flow regimes (laminar, turbulent, transient)?
• RQ2 How do computational costs and memory requirements vary among the eight methods, especially for large datasets?
• RQ3 To what extent do different amplitude computation strategies (e.g., Eq. 37 vs. Eq. 39) affect the accuracy of DMD and HODMD reconstructions?
• RQ4 How well do extended methods like SPOD, HODMD, mPOD, and mrDMD capture multiscale or transient flow features compared to classical POD and DMD?
• RQ5 What is the relative performance of Resolvent Analysis in identifying coherent structures compared to traditional modal decomposition techniques?

Key findings

• HODMD achieved the lowest reconstruction error of $5.85 \times 10^{-13}$ using Eq. (39) for amplitude computation in the 2D laminar wake case, significantly outperforming other methods.
• The method based on full matrix projection (Eq. 39) reduced reconstruction error by six orders of magnitude compared to the snapshot-based approach (Eq. 37), which had an error of $1.64 \times 10^{-6}$.
• SPOD and HODMD demonstrated superior robustness in capturing transient and multiscale features in the 3D cylinder wake case, particularly in the presence of limited data.
• The computational time for the most accurate amplitude computation (Eq. 39) was $6.53 \times 10^{-4}$ seconds, showing that high accuracy can be achieved with minimal overhead.
• The growth-rate-based criterion (Eq. 46) successfully identified dominant, long-lived modes in the turbulent channel flow, even when amplitudes were moderate.
• Resolvent Analysis showed strong similarity to DMD and POD modes, particularly in identifying coherent structures in the turbulent jet, though with higher computational cost than standard DMD.