Manifold Learning for Data-Driven Dynamical System Analysis

Shnitzer, Tal; Talmon, Ronen; Slotine, Jean‐Jacques E.

doi:10.1007/978-3-030-35713-9_14

Cited by 4 publications

(8 citation statements)

References 17 publications

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“…Our method reconciles approaches that rely purely on the temporal structure of the observations with those that approximate the invariant density and ignore the temporal order of measurements. With the recent development of the field of geometric statistics [50,51], and the surge of interest on the concept of manifold hypothesis [52,53], i.e., the consideration that often the state of multi-dimensional dynamical systems is confined on low dimensional regions of the state space, several inference methods have tried to merge geometric and temporal perspectives for identification of stochastic systems. In the Langevin regression framework [54], Callaham et al compute the Kramers-Moyal coefficients, and account for misestimation due to low sampling rate by solving the adjoint Fokker-Planck equation for the coefficients as proposed by Lade [55].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems

Maoutsa¹

2023

Preprint

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Stochastic evolution equations describing the dynamics of systems under the influence of both deterministic and stochastic forces are prevalent in all fields of science. Yet, identifying these systems from sparse-in-time observations remains still a challenging endeavour. Existing approaches focus either on the temporal structure of the observations by relying on conditional expectations, discarding thereby information ingrained in the geometry of the system's invariant density; or employ geometric approximations of the invariant density, which are nevertheless restricted to systems with conservative forces. Here we propose a method that reconciles these two paradigms. We introduce a new data-driven path augmentation scheme that takes the local observation geometry into account. By employing non-parametric inference on the augmented paths, we can efficiently identify the deterministic driving forces of the underlying system for systems observed at low sampling rates. * https://dimitra-maoutsa.gitlab.io/ 2 Here we employ the term path augmentation to refer to what is widely known as data augmentation. We resorted to this term because we consider it as more elegant and better descriptive of the proposed augmentation process, while the term 'data augmentation' is considerably vague.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems

Maoutsa¹

2023

Preprint

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“…Therefore, we should take into account not only the dynamics of the observed variables, but also its interaction with the unobserved variables. This assumption leads to a more complicated problem than (1) and in further section this problem is derived and discussed in detail.…”

Section: Mori-zwanzig Decomposition As a Nonlinear Extension Of Dynam...mentioning

confidence: 99%

“…We denote the result of such simulation as Measurement and plot it in Figure 1. Based on this simulation we construct matrices X + and X − from the DMD problem statement and solve problem (1). Then, we use the obtained operator A dmd to build the reconstructed dynamics according to (19), where the spectrum and eigenvectors of A dmd are used instead of averaged ones.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction

Katrutsa¹,

Utyuzhnikov²,

Oseledets³

2022

Preprint

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The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.

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“…But it is hard to find the topology structure of the Bayesian network and calculate the transfer probability among nodes in the graphical model. The manifold learning models infer low-dimensional manifold of original data based on neighborhood information by non-parametric approaches [15,22]. The models own a solid theoretical basis, but the resolution process needs much time.…”

Section: Related Workmentioning

confidence: 99%

Reimagining City Configuration: Automated Urban Planning via Adversarial Learning

Wang,

Fu,

Wang

et al. 2020

Preprint

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Urban planning refers to the efforts of designing land-use configurations. Effective urban planning can help to mitigate the operational and social vulnerability of a urban system, such as high tax, crimes, traffic congestion and accidents, pollution, depression, and anxiety. Due to the high complexity of urban systems, such tasks are mostly completed by professional planners. But, human planners take longer time. The recent advance of deep learning motivates us to ask: can machines learn at a human capability to automatically and quickly calculate land-use configuration, so human planners can finally adjust machine-generated plans for specific needs? To this end, we formulate the automated urban planning problem into a task of learning to configure land-uses, given the surrounding spatial contexts. To set up the task, we define a land-use configuration as a longitude-latitude-channel tensor, where each channel is a category of POIs and the value of an entry is the number of POIs. The objective is then to propose an adversarial learning framework that can automatically generate such tensor for an unplanned area. In particular, we first characterize the contexts of surrounding areas of an unplanned area by learning representations from spatial graphs using geographic and human mobility data. Second, we combine each unplanned area and its surrounding context representation as a tuple, and categorize all the tuples into positive (well-planned areas) and negative samples (poorly-planned areas). Third, we develop an adversarial land-use configuration approach, where the surrounding context representation is fed into a generator to generate a land-use configuration, and a discriminator learns to distinguish among positive and negative samples. Finally, we devise two new measurements to evaluate the quality of land-use configurations and present extensive experiment and visualization results to demonstrate the effectiveness of our method.

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Manifold Learning for Data-Driven Dynamical System Analysis

Cited by 4 publications

References 17 publications

Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems

Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems

Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction

Reimagining City Configuration: Automated Urban Planning via Adversarial Learning

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