Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries

Drawing upon the bursting mechanism in slow-fast systems, we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates. The indicators are associated with functionals defined in terms of Optimally Time Dependent (OTD) modes. One such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes. In contrast to other choices of subspaces, the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful. We illustrate the application of these indicators on three examples: a prototype low-dimensional model, a body forced turbulent fluid flow, and a unidirectional model of nonlinear water waves. We use Bayesian statistics to quantify the predictive power of the proposed indicators.

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“…One can show, from the Navier-Stokes equation (14), that these three quantities satisfyĖ = I − D along any trajectory.…”

Section: A Governing Equations and Preliminariesmentioning

confidence: 99%

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Farazmand

Sapsis

2016

Phys. Rev. E

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“…We distinguish between approaches that solely rely on pre-computed quantities for the adaptation and approaches that adapt the reduced model from new data that are generated during the online phase. Interpolation between reduced operators and reduced models [2,18,39,51], localization approaches [3,9,11,[19][20][21]36,40,46], and dictionary approaches [30,35] rely on pre-computed quantities but do not incorporate information from new data into the reduced model online. In [4], local reduced models are adapted from partial data online to smooth the transition between the local models.…”

Section: Introductionmentioning

confidence: 99%

Dynamic data-driven model reduction: adapting reduced models from incomplete data

Peherstorfer

Willcox

2016

Adv. Model. and Simul. in Eng. Sci.

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This work presents a data-driven online adaptive model reduction approach for systems that undergo dynamic changes. Classical model reduction constructs a reduced model of a large-scale system in an offline phase and then keeps the reduced model unchanged during the evaluations in an online phase; however, if the system changes online, the reduced model may fail to predict the behavior of the changed system. Rebuilding the reduced model from scratch is often too expensive in time-critical and real-time environments. We introduce a dynamic data-driven adaptation approach that adapts the reduced model from incomplete sensor data obtained from the system during the online computations. The updates to the reduced models are derived directly from the incomplete data, without recourse to the full model. Our adaptivity approach approximates the missing values in the incomplete sensor data with gappy proper orthogonal decomposition. These approximate data are then used to derive low-rank updates to the reduced basis and the reduced operators. In our numerical examples, incomplete data with 30-40 % known values are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.

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“…More recently, the low-rank POD approximations were combined with sparse representation to enable ROM computations with very limited sensors [5,8]. Significant effort has been applied to develop principled, rather than random, sensor placement for ROMs [59,63,15,51]. Although none of the above methods solve the typically NP-hard sensor placement optimization problem, they have been demonstrated to be quite effective in many applications.…”

Section: Related Workmentioning

confidence: 99%

Sparse Sensor Placement Optimization for Classification

Brunton¹,

Brunton²,

Proctor³

et al. 2016

SIAM J. Appl. Math.

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Abstract. Choosing a limited set of sensor locations to characterize or classify a high-dimensional system is an important challenge in engineering design. Traditionally, optimizing the sensor locations involves a brute-force, combinatorial search, which is NP-hard and is computationally intractable for even moderately large problems. Using recent advances in sparsity-promoting techniques, we present a novel algorithm to solve this sparse sensor placement optimization for classification (SSPOC) that exploits low-dimensional structure exhibited by many high-dimensional systems. Our approach is inspired by compressed sensing, a framework that reconstructs data from few measurements. If only classification is required, reconstruction can be circumvented and the measurements needed are orders-of-magnitude fewer still. Our algorithm solves an 1 minimization to find the fewest nonzero entries of the full measurement vector that exactly reconstruct the discriminant vector in feature space; these entries represent sensor locations that best inform the decision task. We demonstrate the SSPOC algorithm on five classification tasks, using datasets from a diverse set of examples, including physical dynamical systems, image recognition, and microarray cancer identification. Once training identifies sensor locations, data taken at these locations forms a low-dimensional measurement space, and we perform computationally efficient classification with accuracy approaching that of classification using full-state data. The algorithm also works when trained on heavily subsampled data, eliminating the need for unrealistic full-state training data.

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Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries

Cited by 58 publications

References 48 publications

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Dynamic data-driven model reduction: adapting reduced models from incomplete data

Sparse Sensor Placement Optimization for Classification

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