scholarly journals Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Zhe Bai ◽  
Liqian Peng
Keyword(s):  
Machine Learning ◽  
Nonlinear Dynamical Systems ◽  
Reduced Order Models ◽  
Simulation Code ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Regression Techniques ◽  
Low Dimensional ◽  
Modern Machine

AbstractAlthough projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

scholarly journals Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

2021 ◽  
Author(s):  
Zhe Bai ◽  
Liqian Peng
Keyword(s):  
Machine Learning ◽  
Nonlinear Dynamical Systems ◽  
Reduced Order Models ◽  
Simulation Code ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Regression Techniques ◽  
Low Dimensional ◽  
Modern Machine

Abstract Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $10^3\times$ in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs.

Multiscale, Multiphenomena Modeling and Simulation at the Nanoscale: On Constructing Reduced-Order Models for Nonlinear Dynamical Systems With Many Degrees-of-Freedom

2003 ◽  
Vol 70 (3) ◽  
pp. 328-338 ◽  
Author(s):  
E. H. Dowell ◽  
D. Tang
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Total System ◽  
Reduced Order Models ◽  
Efficient Computation ◽  
Linear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Value Decomposition

The large number of degrees-of-freedom of finite difference, finite element, or molecular dynamics models for complex systems is often a significant barrier to both efficient computation and increased understanding of the relevant phenomena. Thus there is a benefit to constructing reduced-order models with many fewer degrees-of-freedom that retain the same accuracy as the original model. Constructing reduced-order models for linear dynamical systems relies substantially on the existence of global modes such as eigenmodes where a relatively small number of these modes may be sufficient to describe the response of the total system. For systems with very many degrees-of-freedom that arise from spatial discretization of partial differential equation models, computing the eigenmodes themselves may be the major challenge. In such cases the use of alternative modal models based upon proper orthogonal decomposition or singular value decomposition have proven very useful. In the present paper another facet of reduced-order modeling is examined, i.e., the effects of “local” nonlinearity at the nanoscale. The focus is on nanoscale devices where it will be shown that a combination of global modal and local discrete coordinates may be most effective in constructing reduced-order models from both a conceptual and computational perspective. Such reduced-order models offer the possibility of reducing computational model size and cost by several orders of magnitude.

Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

2010 ◽  
Vol 114 (1160) ◽  
pp. 637-650 ◽  
Author(s):  
M. Arnst ◽  
R. Ghanem ◽  
S. Masri
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Quantitative Description ◽  
Polynomial Expansion ◽  
Reduced Order Models ◽  
State Variables ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Entropy Maximization ◽  
Reduced Order

AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

scholarly journals Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network

2019 ◽  
Author(s):  
Xuping Xie ◽  
Guannan Zhang ◽  
Clayton G. Webster
Keyword(s):  
Differential Operators ◽  
Nonlinear Dynamical Systems ◽  
Dimensional Space ◽  
Fluid Dynamic ◽  
Data Driven ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Highly Nonlinear ◽  
Low Dimensional

In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams-Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches

scholarly journals Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 757 ◽  
Author(s):  
Xuping Xie ◽  
Guannan Zhang ◽  
Clayton G. Webster
Keyword(s):  
Differential Operators ◽  
Nonlinear Dynamical Systems ◽  
Dimensional Space ◽  
Fluid Dynamic ◽  
Data Driven ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Highly Nonlinear ◽  
Low Dimensional

In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams–Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches.

scholarly journals GENETIC ALGORITHM-BASED CALIBRATION OF REDUCED ORDER GALERKIN MODELS

2011 ◽  
Vol 16 (1) ◽  
pp. 233-247 ◽  
Author(s):  
Witold Stankiewicz ◽  
Robert Roszaka ◽  
Marek Morzyńskia
Keyword(s):  
Genetic Algorithm ◽  
Bluff Body ◽  
Calibration Procedure ◽  
Galerkin Projection ◽  
Reduced Order Models ◽  
Data Set ◽  
Reduced Order ◽  
Low Dimensional ◽  
Fluid Behaviour ◽  
Proper Orthogonal

Low-dimensional models, allowing quick prediction of fluid behaviour, are key enablers of closed-loop flow control. Reduction of the model's dimension and inconsistency of high-fidelity data set and the reduced-order formulation lead to the decrease of accuracy. The quality of Reduced-Order Models might be improved by a calibration procedure. It leads to global optimization problem which consist in minimizing objective function like the prediction error of the model. In this paper, Reduced-Order Models of an incompressible flow around a bluff body are constructed, basing on Galerkin Projection of governing equations onto a space spanned by the most dominant eigenmodes of the Proper Orthogonal Decomposition (POD). Calibration of such models is done by adding to Galerkin System some linear and quadratic terms, which coefficients are estimated using Genetic Algorithm.

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