Bulletin of the American Physical Society
75th Annual Meeting of the Division of Fluid Dynamics
Volume 67, Number 19
Sunday–Tuesday, November 20–22, 2022; Indiana Convention Center, Indianapolis, Indiana.
Session L21: Nonlinear Dynamics: Reduced-Order Modeling II |
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Chair: Aaron Towne, University of Michigan Room: 207 |
Monday, November 21, 2022 8:00AM - 8:13AM Author not Attending |
L21.00001: Combining probabilistic and trajectory-based learning for reduced-order modeling Peter J Baddoo, Benjamin Herrmann, Beverley J McKeon, J. Nathan Kutz, Steven L Brunton Learning efficient and accurate reduced-order models is a primary aim of the fluid dynamics community. Recently, data-driven techniques have achieved impressive results in extracting models from data measurements. These techniques usually aim to match the predicted trajectory of the learned model with data measurements. However, the chaotic systems frequently encountered in fluid dynamics are highly sensitive to their parameters, which can prevent data-driven methods from converging accurately. To ameliorate this issue, we suggest a reduced-order modeling framework that blends trajectory-based learning with probability density function matching. Specifically, we optimize our model by using adjoint methods to evaluate the sensitivity of the Kullback–Leibler divergence of the learned model with respect to the measured data. We explore our approach on a range of fluid mechanics problems. |
Monday, November 21, 2022 8:13AM - 8:26AM |
L21.00002: Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance Samuel E Otto, Alberto Padovan, Clarence W Rowley Data-driven reduced-order models often fail to accurately forecast high-dimensional nonlinear systems that are sensitive along coordinates with low-variance. |
Monday, November 21, 2022 8:26AM - 8:39AM |
L21.00003: On-the-fly Reduced-Order Modeling of Transient Flow Response Subject to High-Dimensional External Forcing Alireza Amiri-Margavi, Hessam Babaee |
Monday, November 21, 2022 8:39AM - 8:52AM |
L21.00004: A fast method for computing nonlinear reduced-order models with conserved quantities Mohammad M Farazmand, William Anderson, Zack Hilliard Reduced-order nonlinear solutions (RONS) was developed recently as a powerful method for reduced-order modeling of PDEs. RONS significantly broadens the scope of reduced-order modeling compared to previous linear methods, e.g., Galerkin-type projections. RONS also allows for enforcing conserved quantities of the PDE in the reduced model. In this talk, I will discuss a fast and accurate method for forming the RONS equations. This method exploits the structure of the metric tensor involved in RONS to reduce the computational cost by several orders of magnitude. This speed up allows us to go beyond reduced-order modeling and use RONS for accurate numerical simulation of PDEs. I demonstrate the application of the algorithm on several examples including vortex dynamics in ideal fluids (Euler equation) and turbulence (Navier-Stokes equation). In particular, I will discuss the effect of enforcing conserved quantities on energy and enstrophy cascades. |
Monday, November 21, 2022 8:52AM - 9:05AM |
L21.00005: Data-driven reduced-order models using quadratic manifolds Aniketh Kalur, Rudy Geelen, Karen E Willcox Data-driven methods such as dynamic mode decomposition (DMD) provide linear reduced-order models (ROMs) to approximate nonlinear systems. Typically the DMD-based ROMs perform well on a training data set; however, the ROMs' predictive accuracy is relatively poor, especially for complex nonlinear systems. In this talk, we propose a data-driven quadratic closure term for the DMD-based ROM, such that the resulting ROM has a linear-quadratic structure. Using the proposed closure term leads to a ROM whose trajectories evolve on a quadratic manifold; this is an improvement over the DMD-based ROM whose reduced-order trajectories evolve in a linear subspace. The quadratic manifold ROM display improved predictions and effectively approximates shocks and transient behavior. We demonstrate the enhancement of prediction capabilities on problems of traveling waves, the cylinder flow, and the lid-driven cavity. We also use the example of a traveling detonation wave in a shock tube to illustrate the ability of the proposed ROM to approximate shocks. |
Monday, November 21, 2022 9:05AM - 9:18AM |
L21.00006: Nonlinear Oblique Projections for Reduced-Order Modeling using Constrained Autoencoders Gregory R Macchio, Samuel E Otto, Clarence W Rowley Reduced-order modeling techniques based on linear projections are known to provide inefficient dimensionality reduction for systems that evolve near highly curved nonlinear manifolds. We introduce nonlinear projections parameterized by autoencoders, constrained so that composing of the encoder with the decoder is the identity. In existing techniques, the full-order model's time derivative is orthogonally projected onto the tangent space of the nonlinear manifold and is defined only by decoder. In this work, we argue the direction of projection is also important in capturing the dynamics. This is especially true in non-normal systems such as shear-dominated fluid flows. To address this problem, we additionally use the encoder and its derivative to determine the direction of projection which in general may be oblique. Furthermore, we optimize the network using cost functions that balance the reconstruction accuracy of states via the autoencoder against the sensitivity of future outputs of the system to state perturbations. We demonstrate this method on several examples, including a three-dimensional model of flow past a circular cylinder and the complex Ginzburg-Landau equation. |
Monday, November 21, 2022 9:18AM - 9:31AM |
L21.00007: SPOD-based Spectral Galerkin Reduced Order Models Cong Lin, Oliver T. Schmidt Spectral proper orthogonal decomposition (SPOD) is a space-time-optimal modal decomposition that extracts orthogonal, single-frequency coherent structures from data and is applicable to ergodic processes like many turbulent flows. Recently, new classes of frequency domain reduced-order models (ROMs) have been introduced by Lin et. al. (2019) and Towne (2021). They rely on a space-time Galerkin projection onto the SPOD modes, which makes use of the favorable properties of the decomposition. The proposed method is based on the orthogonal projection of the governing equations in the frequency domain onto an SPOD basis to compress the system at each discrete frequency. Thus, the resulting ROM is a system of algebraic equations that are solved at each frequency. An inverse Fourier transform of the spectral solution yields the entire time-domain solution over any finite time horizon of interest – no time integration is necessary. The model reduction process is applied to the Navier-Stokes equations under nonlinear colored forcing from the Reynolds-stresses for the example of a chaotic lid-driven cavity flow at Reynolds numbers of 15000 and higher. We also illustrate the explicit inclusion of triadic interactions via convolution with a linear time-varying system (bilinear convection) and elaborate the generalization to a fully quadratic nonlinearity. |
Monday, November 21, 2022 9:31AM - 9:44AM |
L21.00008: On-the-fly sparse interpolation for stochastic reduced-order modeling with time-dependent bases Mohammad Hossein Naderi, Hessam Babaee, Mohammad Hossein Naderi One of the main challenges in reduced-order modeling of stochastic partial differential equations (SPDEs) with time-dependent bases (TDBs) is the presence of non-polynomial nonlinear terms which require the same computational cost as solving full-order models. This work proposes an adaptive sparse interpolation algorithm that enables stochastic reduced-order modeling based on TDBs with a large number of samples by reducing the computational cost of solving TDBs evolution equations. There is no need to perform any offline computations with this approach and it can adapt on-the-fly to any transient changes in dynamics. The presented method constructs a low-rank approximation for the right-hand side of the SPDE using the discrete empirical interpolation method (DEIM) with a rank-adaptive strategy to control the error of the sparse interpolation. Our method achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers' equation and (2) two-dimensional compressible Navier-Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, computational costs have been reduced by orders of magnitude. |
Monday, November 21, 2022 9:44AM - 9:57AM |
L21.00009: Data-driven mean-field modeling for complex wake dynamics -- Towards automatable data-driven ROM Nan DENG, Luc Pastur, Marek Morzynski, Bernd R Noack Increasing high-quality flow data are generated by simulations and experiments with advanced measurement technology and high-performance computers. Data-driven methods and machine learning (ML) techniques are thus bringing new opportunities and challenges to the current research paradigm in the era of big data. |
Monday, November 21, 2022 9:57AM - 10:10AM |
L21.00010: Time-Series Velocity Field Reconstruction Based on Unsteady Pressure Sensors for Feedback Control of Separated Flow around an Airfoil Yoshiki Anzai, Shintaro Goto, Yasuo Sasaki, Kumi Nakai, Atsushi Komuro, Taku Nonomura A low-dimensional flow field is estimated from unsteady pressure sensors data that can be attached to the airfoil surface, for feedback control of the flow field around an airfoil. The method estimates a low-dimensional flow field represented by proper orthogonal decomposition (POD) modes with a Kalman filter. In this study, we aim to improve the estimation accuracy of the method based on the Kalman filter and to investigate the effects of the Kalman filter hyperparameters, the number of POD modes to be estimated, and the time delay of the observations on the accuracy of the flow field estimation, based on wind tunnel test data. We also propose a nonlinear state-space model as a state-space representation, which is an improvement of the conventional linear state-space model, and perform estimation using an extended Kalman filter, and investigate the estimation accuracy in the same way. The analysis showed that the coefficients of the observation error covariance matrix, which is a hyperparameter of the Kalman filter, and the number of POD modes to be estimated contribute to the accuracy of the estimation. The consideration of the time delay of the observations did not improve the accuracy of the estimation. |
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