Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (i) a direct correction of the identified bias using known noise properties, (ii) combining the results of performing DMD forwards and backwards in time, and (iii) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm, and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in recent literature.

The Dynamic Mode Decomposition (DMD)-a popular method for performing data-driven Koopman spectral analysis-has gained increased adoption as a technique for extracting dynamically meaningful spatiotemporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD modelforecasts. Despite its widespread use and utility, DMD regularly fails to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD's first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD's performance is demonstrated in numerical and experimental fluids examples.

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We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard "batch-processed" formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments.Dynamic mode decomposition (DMD) is a data-driven computational technique capable of extracting dynamical information from flowfields measured in physical experiments or generated by direct numerical simulations. 1 Since its introduction in 2008, 2 DMD has been used in the analysis of numerous fluid mechanical systems (e.g., bluff body flows, 3 jet flows, 4,5 and viscoelastic fluid flows 6 ) and has gained increasing popularity owing to its ability to reveal and quantify the dynamics of a flow, even when those dynamics are nonlinear. 4,7 DMD operates on snapshots of the flowfield (e.g., velocity, vorticity, pressure) and their time-shifted counterparts-obtained either from experiments or numerical simulations-to compute the eigenvalues ("DMD eigenvalues") and eigenvectors ("DMD modes") of a linear operator that best fits the associated dynamics in a least-squares sense. The DMD modes represent spatial fields that often highlight coherent structures in the flow, while the associated DMD eigenvalues dictate the decay/growth rates and oscillation frequencies of these modes. As such, access to DMD modes and eigenvalues enables a reconstruction of the dynamics associated with a given flowfield. Other modal decomposition techniques, such as the commonly employed proper orthogonal decomposition (POD), only compute spatial modes associated with the flow. 8 Although spatial modes can offer valuable information regarding coherent structures and other flow qualities (e.g., in the case of POD, they determine the most energetic modes), characterizing the underlying dynamics relies upon projecting these spatial modes onto an assumed dynamical form. DMD offers an advantage over these other modal decomposition techniques in that it computes both spatial modes and their associated temporal behaviors, thus removing any guesswork associated with realizing a dynamical representation of the system.To date, researchers have viewed DMD as a post-processing tool; that is, a method that requires the entire experimental or computational dataset to be available prior to commencing

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Machine learning (i.e., modern data-driven optimization and applied regression) is a rapidly growing field of research that is having a profound impact across many fields of science and engineering. In the past decade, machine learning has become a critical complement to existing experimental, computational, and theoretical aspects of fluid dynamics. In this short article, we are excited to introduce this special issue highlighting a number of promising avenues of ongoing research to integrate machine learning and data-driven techniques in the field of fluid dynamics. We will also attempt to provide a broader perspective, outlining recent successes, opportunities, and open challenges, while balancing optimism and skepticism. In the field of fluid dynamics, there is an interesting parallel between the rise of machine learning in recent years and the rise of computational science decades earlier. Neither approach fundamentally changes the scientific questions being asked nor the higher-level objectives. Rather, both approaches provide sophisticated tools for analysis based on emerging technologies, enabling the community to address scientific questions at a greater scale and a broader scope than was previously possible. In the early years of computational fluid dynamics, there were voices of both extreme skepticism and open-ended optimism that these new approaches would supplant existing techniques. In reality, computational techniques have provided another valuable perspective for scientific inquiry, complementing more traditional approaches. It is therefore reasonable to believe that machine learning and data-intensive analysis will have a similar impact, complementing other well-established techniques to expand our collective capabilities. Machine learning offers a wealth of techniques to discover patterns in high-dimensional data [5,6], extending traditional modal expansions that have been a cornerstone of fluid dynamics for decades [26,27]. Despite this great potential, it is important to recognize that these algorithms must be used properly, and that a single tool alone will not be equipped to address every task. The same factors that make data-driven and machine learning methods so appealing-namely, that they are relatively easy to use and do not require expert knowledge-can also serve as a potential downfall. Much as users of CFD software are cautioned against blind application of these numerical tools without proper knowledge, training, and verification/validation, we must adopt a similar philosophy for the use of data-driven and machine learning tools. Communicated by Tim Colonius.

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