A deep learning enabler for nonintrusive reduced order modeling of fluid flows

Pawar, Suraj; Rahman, Sk. Mashfiqur; Vaddireddy, Harsha; San, Omer; Rasheed, Adil; Vedula, Prakash

doi:10.1063/1.5113494

Cited by 148 publications

(62 citation statements)

References 136 publications

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“…16. We note that at time t = 2 the exact solution develops a relatively sharp (albeit still smooth) transition layer at this relatively low visocity.…”

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confidence: 66%

Data-driven deep learning of partial differential equations in modal space

Xiu

2020

Journal of Computational Physics

125

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We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.ing studies are relatively limited, as they mostly focus on learning certain types of PDE or identifying the exact terms in the PDE from a (large) dictionary of possible terms. The specific novelty of this paper is that the proposed method seeks to recover/approximate the evolution operator of the underlying unknown PDE and is applicable for general class of PDEs. The evolution operator completely characterizes the time evolution of the solution. Its recovery allows one to conduct prediction of the underlying PDE and is effectively equivalent to the recovery of the equation. This is an extension of the equation recovery work from [17], where the flow map of the underlying unknown dynamical system is the goal of recovery. Unlike the ODE systems considered in [17], PDE systems, which is the focus of this paper, are of infinite dimension. In order to cope with infinite dimension, our method first reduces the problem into finite dimensions by utilizing a properly chosen modal space, i.e., generalized Fourier space. The equation recovery task is then transformed into recovery of the generalized Fourier coefficients, which follow a finite dimensional dynamical system. The approximation of the finite dimensional evolution operator of the reduced system is then carried out by using deep neural network, particularly the residual network (ResNet) which has been shown to be particularly suitable for this task [17]. One of the advantages of the proposed method is that, by focusing on evolution operator, it eliminates the need for time derivatives data of the state variables. Time derivative data, often required by many existing methods, are difficult to acquire in practice and susceptible to (additional) errors when computed numerically. Moreover, the proposed method can cope with solution data that are more sparsely or unevenly distributed in time. Since the proposed framework is rather general, we present seve...

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“…16. We note that at time t = 2 the exact solution develops a relatively sharp (albeit still smooth) transition layer at this relatively low visocity.…”

mentioning

confidence: 66%

Data-driven deep learning of partial differential equations in modal space

Xiu

2020

Journal of Computational Physics

125

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“…Reduced basis method for optimal control of unsteady viscous flows. International Journal of Computational Fluid Dynamics, 15(2): [97][98][99][100][101][102][103][104][105][106][107][108][109][110][111][112][113]2001. Figure 31: The first local basis function for vorticity field from PID application over the first subinterval (i.e., for t κ (Np−1) ≤ t ≤ 40) for double shear layer problem using different number of intervals.…”

Section: Boussinesq Problemmentioning

confidence: 99%

Memory embedded non-intrusive reduced order modeling of non-ergodic flows

et al. 2019

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Generating a digital twin of any complex system requires modeling and computational approaches that are efficient, accurate, and modular. Traditional reduced order modeling techniques are targeted at only the first two but the novel non-intrusive approach presented in this study is an attempt at taking all three into account effectively compared to their traditional counterparts. Based on dimensionality reduction using proper orthogonal decomposition (POD), we introduce a long short-term memory (LSTM) neural network architecture together with a principal interval decomposition (PID) framework as an enabler to account for localized modal deformation, which is a key element in accurate reduced order modeling of convective flows. Our applications for convection dominated systems governed by Burgers, Navier-Stokes, and Boussinesq equations demonstrate that the proposed approach yields significantly more accurate predictions than the POD-Galerkin method, and could be a key enabler towards near real-time predictions of unsteady flows. and perhaps much beyond. With the recent wave of digitization, reduced order modeling can be viewed as one of the key enablers to bring the promise of the digital twinning concept closer to reality [38]. Therefore, there is a continuous demand for the development of accurate reduced order models for complex physical phenomena. In projection-based ROMs, the most widely used technique, the discrete high-dimensional operators are projected onto a lower-dimensional space, so that the problem can be solved more efficiently in this reduced space [39][40][41][42][43][44].One of the very early-developed and well-known approaches to build this reduced space is Fourier analysis. However, it assumes universal basis functions (or modes) which have no specific relation to the physical system. On the other hand, snapshot-based model reduction techniques tailor a reduced space that best fits the problem by extracting the underlying coherent structures that controls the major dynamical evolution we are interested in. Proper orthogonal decomposition (POD) is a very popular and well-established approach extracting the modes which most contributes to the total variance [45,46]. In fluid dynamics applications, where we are mostly interested in the velocity field, those modes contain the largest amount of kinetic energy [47,48]. That is why POD is usually classified as an energy-based decomposition method. Another popular approach for model order reduction is the dynamic mode decomposition (DMD) [49][50][51][52][53][54] which generates a number of modes, each characterized by an oscillating frequency and growth/decay rate. In the present study, we are interested in the application of POD for dimensionality reduction.POD generates a set of spatial orthonormal basis functions, each containing a significant amount of total energy. To obtain a reduced representation of a system, the first few modes are selected, and the remaining are truncated assuming their contribution to the system's behavior is minimum (i.e.,...

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“…While the stationary test case is one of the most classical paradigms for wake flows exhibiting large scale vortex shedding (see Williamson (1996) for a review), the transient configuration in turbulent conditions has received considerably less attention. Previous investigations on data-driven decomposition of the transient cylinder focus on the onset of the vortex shedding, both for reduced-order modeling (Murata et al, 2019, Pawar et al, 2019, Siegel et al, 2008 and flow control applications (Bergmann and Cordier, 2008, Gronskis et al, 2009, Rabault et al, 2019.…”

Section: Introductionmentioning

confidence: 99%

Multiscale proper orthogonal decomposition (mPOD) of TR-PIV data—a case study on stationary and transient cylinder wake flows

Mendez

Hess

Watz

et al. 2020

Meas. Sci. Technol.

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Data-driven decompositions of Particle Image Velocimetry (PIV) measurements are widely used for a variety of purposes, including the detection of coherent features (e.g., vortical structures), filtering operations (e.g., outlier removal or random noise mitigation), data reduction and compression. This work presents the application of a novel decomposition method, referred to as Multiscale Proper Orthogonal Decomposition (mPOD, Mendez et al 2019) to Time-Resolved PIV (TR-PIV) measurement. This method combines Multiresolution Analysis (MRA) and standard Proper Orthogonal Decomposition (POD) to achieve a compromise between decomposition convergence and spectral purity of the resulting modes. The selected test case is the flow past a cylinder in both stationary and transient conditions, producing a frequency-varying Karman vortex street. The results of the mPOD are compared to the standard POD, the Discrete Fourier Transform (DFT) and the Dynamic Mode Decomposition (DMD). The mPOD is evaluated in terms of decomposition convergence and time-frequency localization of its modes. The multiscale modal analysis allows for revealing beat phenomena in the stationary cylinder wake, due to the three-dimensional nature of the flow, and to correctly identify the transition from various stationary regimes in the transient test case.

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A deep learning enabler for nonintrusive reduced order modeling of fluid flows

Cited by 148 publications

References 136 publications

Data-driven deep learning of partial differential equations in modal space

Data-driven deep learning of partial differential equations in modal space

Memory embedded non-intrusive reduced order modeling of non-ergodic flows

Multiscale proper orthogonal decomposition (mPOD) of TR-PIV data—a case study on stationary and transient cylinder wake flows

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