Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

Wu, Yuqi; Hetmaniuk, Ulrich

doi:10.1002/nme.4883

Cited by 8 publications

(9 citation statements)

References 42 publications

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“…Note that the approximate solution incurs no error if the approximate solutions exactly satisfy the FOM O∆E (2) such that the residual is zero at all time instances. This also illustrates why the residual norm is often viewed as a useful error indicator for guiding greedy methods for snapshot collection [7,6,19,48,49] or trust-region optimization algorithms [52,50,51].…”

Section: A Posteriori Error Boundsmentioning

confidence: 94%

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Parish

Carlberg

2020

Computer Methods in Applied Mechanics and Engineering

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This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref.[15] to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features-which comprise error indicators that are derived from standard a posteriori error-quantification techniques-to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

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Section: A Posteriori Error Boundsmentioning

confidence: 94%

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Parish

Carlberg

2020

Computer Methods in Applied Mechanics and Engineering

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“…, µ K } to be selected either a priori guided by physical intuition, or by sampling techniques like random or latin hypercube (LHS) sampling (see, e.g., [53]) and sparse grids (see, e.g., [54]). However, the offline construction in Algorithm 4 can accommodate more general training techniques such as the greedy [55] and POD-greedy [56] algorithms, possibly combined with adaptivity [57,58], heuristic error indicators [52], and localized bases [59]. To this end, however, suitable a posteriori error estimates are needed.…”

Section: C) Extract Global Solution Matrix and Vector Basesmentioning

confidence: 99%

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

Negri

Manzoni

Amsallem

2015

Journal of Computational Physics

129

106

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In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems.

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“…Many if not all of these a posteriori error indicators rely on the inexpensive computation of the norm of residuals associated with the HDM solution . As these residuals are of large dimension, most of the focus in the context of repeated analyses has been in the case of separable, affine parameter dependency of the HDM .…”

Section: Introductionmentioning

confidence: 99%

“…The two quantities are often found to be highly correlated, and as a result, the norm of the residual is often considered to monitor the convergence of an iterative procedure for solving a given set of linear or nonlinear equations. In the context of model reduction, the norm of the residual is used in greedy approaches as well as determining when a ROM loses accuracy in the context of PDE‐constrained optimization . Here, a linear model is constructed to estimate the true error using the residual norm ( a posteriori error indicator).…”

Section: Introductionmentioning

confidence: 99%

An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

Paul-Dubois-Taine

Amsallem

2014

Numerical Meth Engineering

142

164

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SUMMARYAn adaptive and efficient approach for constructing reduced-order models (ROMs) that are robust to changes in parameters is developed. The approach is based on a greedy sampling of the underlying high-dimensional model (HDM) together with an efficient procedure for exploring the configuration space and identifying parameters for which the error is likely to be high. Because this exploration is based on a surrogate model for an error indicator, it is amenable to a fast training phase. Furthermore, a model for the exact error based on the error indicator is constructed and used to determine when the greedy procedure reaches a desired error tolerance. An efficient procedure for updating the reduced-order basis constructed by proper orthogonal decomposition is also introduced in this paper, avoiding the cost associated with computing largescale singular value decompositions. The proposed procedure is then shown to require less evaluations of a posteriori error estimators than the classical procedure in order to identify locations of the parameter space to be sampled. It is illustrated on the training of parametric ROMs for three linear and nonlinear mechanical systems, including the realistic prediction of the response of a V-hull vehicle to underbody blasts. Copyright

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Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

Cited by 8 publications

References 42 publications

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

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