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Nonlinear Model Order Reduction using Diffeomorphic Transformations of a Space-Time Domain [article]

Hendrik Kleikamp, Mario Ohlberger, Stephan Rave
2022 arXiv   pre-print
In the linear space of velocity fields, standard model order reduction techniques, such as proper orthogonal decomposition, can be applied to extract a reduced subspace.  ...  To this end, nonlinear approaches for model order reduction of hyperbolic conservation laws are required.  ...  The equation of interest reads In this work we describe a new approach for nonlinear model order reduction for parametrized hyperbolic equations.  ... 
arXiv:2203.05833v1 fatcat:tpr2agfc4jeelmvwwrutiohitm

Optimal Combination of EEFs for the Model Reduction of Nonlinear Partial Differential Equations

Jun Shuai, Xuli Han
2013 Journal of Applied Mathematics  
Proper orthogonal decomposition is a popular approach for determining the principal spatial structures from the measured data.  ...  In this paper, an optimal combination of EEFs is proposed for model reduction of nonlinear partial differential equations (PDEs), obtained by the basis function transformation from the initial EEFs.  ...  Acknowledgment Financial support from National Natural Science Foundation of China (11271376) is gratefully acknowledged.  ... 
doi:10.1155/2013/347248 fatcat:gzfmsrghsrbcbk2h53qb337tfy

Digital Twin Data Modelling by Randomized Orthogonal Decomposition and Deep Learning [article]

Diana Alina Bistrian and Omer San and Ionel Michael Navon
2022 arXiv   pre-print
We introduce a novel algorithm that combines the advantages of Krylov based dynamic mode decomposition with proper orthogonal decomposition and outperforms the selection of the most influential modes.  ...  We prove that randomized orthogonal decomposition algorithm provides several advantages over SVD empirical orthogonal decomposition methods and mitigates the projection error formulating a multiobjective  ...  Powerful Tools on ROM and Previous Work Among several model order reduction techniques that performs well with non-intrusive data, Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (  ... 
arXiv:2206.08659v1 fatcat:2eh4a2c4qjawfmgfwuph7wzmvq

Control of Nonlinear PDEs based on Space Vectors Clustering reduced order systems

Samir Sahyoun, Seddik M. Djouadi
2014 IFAC Proceedings Volumes  
Proper Orthogonal Decomposition (POD) is widely used to reduce the order of such systems but it assumes that data belongs to a linear space and therefore fails to capture the nonlinear degrees of freedom  ...  To overcome this problem, we develop a Space Vector Clustering (SVC) POD and use the reduced order model to design the controller which will then be applied to the full order system.  ...  For nonlinear systems, Proper Orthogonal Decomposition (POD) is a model reduction technique that proved efficient performance when used to reduce models that approximate nonlinear infinite dimensional  ... 
doi:10.3182/20140824-6-za-1003.02560 fatcat:hdcowgbddfcd7crzknxnycaxyy

LPV Modelling and Control of Burgers' Equation

Seyed Mahdi Hashemi, Herbert Werner
2011 IFAC Proceedings Volumes  
A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction and the accuracy of the reduced model is verified.  ...  The one-dimensional viscous Burgers' equation is discretized using a finite difference scheme and the boundary conditions are taken as control inputs.  ...  Using proper orthogonal decomposition (POD) to obtain optimal basis functions leads to low-order models which can represent the original system reasonably.  ... 
doi:10.3182/20110828-6-it-1002.03318 fatcat:urnwanxwrffcxjg7b6zv2jr3wu

Suboptimal Feedback Control of Nonlinear Distributed Parameter Systems by Stable Manifold Method

K. Hamaguchi, G. Nishida, N. Sakamoto
2014 IFAC Proceedings Volumes  
We apply this method to a reduced-order system obtained from the proper orthogonal decomposition (POD) and Galerkin projection.  ...  The feasibility of the design is demonstrated by the numerical example of feedback controls of the viscous Burgers' equation.  ...  We solve this problem by using of a unified model reduction by the proper orthogonal decomposition (POD) and Galerkin projection (Holmes et al. (1998) ).  ... 
doi:10.3182/20140824-6-za-1003.02606 fatcat:bswu37sqlrad7azgw5oyyvekca

Evolve Filter Stabilization Reduced-Order Model for Stochastic Burgers Equation

Xuping Xie, Feng Bao, Clayton Webster
2018 Fluids  
The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model  ...  In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems.  ...  order model POD Proper orthogonal decomposition DF Differential filter SBE Stochastic Burgers equation SDE Stochastic differential equation SPDE Stochastic partial differential equation Figure 1 . 1 Reduced  ... 
doi:10.3390/fluids3040084 fatcat:3kfd2hynn5eqhfryrjqm66fxqi

High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics

Diana A. Bistrian
2022 Modelling  
equations onto the reduced modes basis).  ...  This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence  ...  Conflicts of Interest: The author declares no conflict of interest.  ... 
doi:10.3390/modelling3030020 fatcat:ua42evl4k5fnxom5jvplnbwvkm

Density-matrix renormalization for model reduction in nonlinear dynamics

Thorsten Bogner
2007 Physical Review E  
We present a novel approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition (POD).  ...  The efficiency of the algorithm is tested on the one dimensional Burgers equations and a one dimensional equation of the Fisher type as nonlinear model systems.  ...  PROPER ORTHOGONAL DECOMPOSITION The proper orthogonal decomposition is a linear projection method which is widely used in model reduction. On this topic an extensive literature exist.  ... 
doi:10.1103/physreve.76.056707 pmid:18233790 fatcat:m7eosn377fhddontu4op7kkx5a

Nonlinear model reduction for fluid flows

Samir Sahyoun, Seddik M. Djouadi
2011 Proceedings of the 2011 American Control Conference  
For example, proper orthogonal decomposition (POD) fails to capture the nonlinear degrees of freedom in these systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean  ...  Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models.  ...  S Among the multitude of model reduction techniques, the proper orthogonal decomposition (POD) is arguably the most popular one method used in deriving reduced models for fluid flows governed by nonlinear  ... 
doi:10.1109/acc.2011.5991592 fatcat:jb4qcsbklrf7rfcn4q6efki4du

Stability Domains for Quadratic-Bilinear Reduced-Order Models [article]

Boris Kramer
2021 arXiv   pre-print
Numerical results for a LQG-balanced ROM of Burgers' equation, a proper orthogonal decomposition ROM of FitzHugh-Nagumo, and a non-intrusive ROM of Burgers' equation demonstrate the scalability and quantitative  ...  - per definition - be derived from the full-order model.  ...  For general nonlinear systems, proper orthogonal decomposition (POD) [16] and the reduced basis method [15] are the most commonly used model reduction methods.  ... 
arXiv:2009.02769v2 fatcat:uvt5t7gx6jebrdekzwu7cau6o4

Memory embedded non-intrusive reduced order modeling of non-ergodic flows [article]

Shady E. Ahmed, Sk. Mashfiqur Rahman, Omer San, Adil Rasheed, Ionel M. Navon
2019 arXiv   pre-print
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.  ...  , or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.  ... 
arXiv:1910.07649v1 fatcat:7jlfk5nk4baglafsjkaagyzxfu

Boundary Control of the Burgers Equation: Optimality Conditions and Reduced-order Approach [chapter]

Stefan Volkwein
2001 Optimal Control of Complex Structures  
First, the standard reduced order proper orthogonal decomposition model, which has been extracted from the governing equations without control inputs, was evaluated and illustrated the satisfactory results  ...  In this paper, a reduced order model is reconstructed for boundary control and excitation of the unsteady viscous Burgers equation.  ...  Finally, the discretization of the ROM equation, the order reduction manner and the results are discussed in the next sections. 2-Proper Orthogonal Decomposition The POD Reduced-order modeling begins  ... 
doi:10.1007/978-3-0348-8148-7_22 fatcat:hytd3brfxbeybk36iizzguh7ru

Using recurrent neural networks for nonlinear component computation in advection-dominated reduced-order models [article]

Romit Maulik, Vishwas Rao, Sandeep Madireddy, Bethany Lusch, Prasanna Balaprakash
2019 arXiv   pre-print
In this paper, we present a long short-term memory neural network to approximate the nonlinear component of the reduced-order model (ROM) of an advection-dominated partial differential equation.  ...  Our results show that the proposed framework recovers transient dynamics accurately without nonlinear term computations in full-order space and represents a cost-effective alternative to solely equation-based  ...  This research was funded in part and used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.  ... 
arXiv:1909.09144v2 fatcat:ieak4k5b2za7rabvzunddegbai

POD-based feedback control of the burgers equation by solving the evolutionary HJB equation

K. Kunisch, L. Xie
2005 Computers and Mathematics with Applications  
It is based on model reduction by proper orthogonal decomposition combined with efficient numerical methods for solving the resulting low-order evolutionary Hamilton-Jacobi-Bellman (HJB) equation.  ...  The method for solving the HJB equation is first tested on several LD problems and then successfully applied to the control of the reduced order Burgers equation.  ...  Since proper orthogonal decomposition contains an efficient inherent mechanism for choosing the basis elements we use this approach in the present paper for model reduction.  ... 
doi:10.1016/j.camwa.2004.07.022 fatcat:ii74s4iv75hs3ama4dcakcbwl4
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