A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

2021 ◽  
pp. 110907
Author(s):  
Zhen Gao ◽  
Yifan Lin ◽  
Xiang Sun ◽  
Xueying Zeng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nearest Neighbors ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Order Method ◽  
Partial Differential ◽  
K Nearest Neighbors ◽  
Reduced Order ◽  
Mode Decomposition

Modeling the Turbulent Wake Behind a Wall-Mounted Square Cylinder

2020 ◽  
Author(s):  
Christian Amor ◽  
José M Pérez ◽  
Philipp Schlatter ◽  
Ricardo Vinuesa ◽  
Soledad Le Clainche
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Turbulent Wake ◽  
Orthogonal Decomposition ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Square Cylinder ◽  
Complex Flow ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Proper Orthogonal

Abstract This article introduces some soft computing methods generally used for data analysis and flow pattern detection in fluid dynamics. These techniques decompose the original flow field as an expansion of modes, which can be either orthogonal in time (variants of dynamic mode decomposition), or in space (variants of proper orthogonal decomposition) or in time and space (spectral proper orthogonal decomposition), or they can simply be selected using some sophisticated statistical techniques (empirical mode decomposition). The performance of these methods is tested in the turbulent wake of a wall-mounted square cylinder. This highly complex flow is suitable to show the ability of the aforementioned methods to reduce the degrees of freedom of the original data by only retaining the large scales in the flow. The main result is a reduced-order model of the original flow case, based on a low number of modes. A deep discussion is carried out about how to choose the most computationally efficient method to obtain suitable reduced-order models of the flow. The techniques introduced in this article are data-driven methods that could be applied to model any type of non-linear dynamical system, including numerical and experimental databases.

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

Reduced-Order Modeling for Dynamic Mode Decomposition Without an Arbitrary Sparsity Parameter

AIAA Journal ◽  
2020 ◽  
Vol 58 (9) ◽  
pp. 3919-3931 ◽  
Author(s):  
John Graff ◽  
Matthew J. Ringuette ◽  
Tarunraj Singh ◽  
Francis D. Lagor
Keyword(s):  
Reduced Order Modeling ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Reduced Order ◽  
Mode Decomposition

scholarly journals From snapshots to modal expansions – bridging low residuals and pure frequencies

2016 ◽  
Vol 802 ◽  
pp. 1-4 ◽  
Author(s):  
Bernd R. Noack
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Orthogonal Decomposition ◽  
Operating Conditions ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Simple Method ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Optimality Property ◽  
Proper Orthogonal

Data-driven low-order modelling has been enjoying rapid advances in fluid mechanics. Arguably, Sirovich (Q. Appl. Maths, vol. XLV, 1987, pp. 561–571) started these developments with snapshot proper orthogonal decomposition, a particularly simple method. The resulting reduced-order models provide valuable insights into flow physics, allow inexpensive explorations of dynamics and operating conditions, and enable model-based control design. A winning argument for proper orthogonal decomposition (POD) is the optimality property, i.e. the guarantee of the least residual for a given number of modes. The price is unpleasant frequency mixing in the modes which complicates their physical interpretation. In contrast, temporal Fourier modes and dynamic mode decomposition (DMD) provide pure frequency dynamics but lose the orthonormality and optimality property of POD. Sieber et al. (J. Fluid Mech., vol. 792, 2016, pp. 798–828) bridge the least residual and pure frequency behaviour with an ingenious interpolation, called spectral proper orthogonal decomposition (SPOD). This article puts the achievement of the TU Berlin authors in perspective, illustrating the potential of SPOD and the challenges ahead.

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