Reconstruction of normal forms by learning informed observation geometries from data

Yair, Or; Talmon, Ronen; Coifman, Ronald R.; Kevrekidis, Ioannis G.

doi:10.1073/pnas.1620045114

Cited by 68 publications

(74 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In such cases, the patterns recovered by VSA naturally factor out data redundancies, which generally enhances both robustness and physical interpretability of the results. Besides spatiotemporal data, the bundle construction described above may be useful in other scenarios, e.g., analysis of data generated by dynamical systems with varying parameters [46].…”

Section: Bundle Structure Of Spatiotemporal Datamentioning

confidence: 99%

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

et al. 2019

View full text Add to dashboard Cite

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

show abstract

Section: Bundle Structure Of Spatiotemporal Datamentioning

confidence: 99%

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Second, following the step of data fusion, one can attempt to model the observed multivariable dynamics. Here one can employ several modeling methodologies, from mechanistic modeling of specific molecular and tissue-level processes [31][32][33][34][35], to equationfree approaches, which aim to deduce the underlying mechanisms directly from data [36,37].…”

Section: Discussionmentioning

confidence: 99%

Synthesizing developmental trajectories

Villoutreix

Andén

Lim

et al. 2017

Preprint

Self Cite

View full text Add to dashboard Cite

Dynamical processes in biology are studied using an ever-increasing number of techniques, each of which brings out unique features of the system. One of the current challenges is to develop systematic approaches for fusing heterogeneous datasets into an integrated view of multivariable dynamics. We demonstrate that heterogeneous data fusion can be successfully implemented within a semi-supervised learning framework that exploits the intrinsic geometry of high-dimensional datasets. We illustrate our approach using a dataset from studies of pattern formation in Drosophila. The result is a continuous trajectory that reveals the joint dynamics of gene expression, subcellular protein localization, protein phosphorylation, and tissue morphogenesis. Our approach can be readily adapted to other imaging modalities and forms a starting point for further steps of data analytics and modeling of biological dynamics. Author summaryA wide range of problems in biology require analysis of multivariable dynamics in space and time. As a rule, the multiscale nature and complexity of real systems precludes simultaneous monitoring of all the relevant variables, and multivariable dynamics must be synthesized from partial views provided by different experimental techniques. We present a formal framework for accomplishing this task in the context of imaging studies of pattern formation in developing tissues.

show abstract

“…When using such basis functions in the computation of inner products, as in (4), this choice harnesses the average of observations with a similar underlying character (in the sense that their initial conditions and time indices are similar) to the formulation of the informed metric. The justification for choosing this type of basis function is further discussed in the references ,,,. Other choices of basis functions are possible and may be more effective; we are currently exploring using the eigenvectors of diffusion maps as our basis functions.…”

Section: Methodsmentioning

confidence: 99%

“…The paper is organized as follows: Section 2 starts with a reasonably self‐contained description of the important components of the approach introduced in Yair et al . for data driven construction of parameter space and state space realizations.…”

Section: Introductionmentioning

confidence: 99%

Data‐driven Evolution Equation Reconstruction for Parameter‐Dependent Nonlinear Dynamical Systems

Sroczynski

Yair

Talmon

et al. 2018

Israel Journal of Chemistry

Self Cite

View full text Add to dashboard Cite

When studying observations of chemical reaction dynamics, closed form equations based on a putative mechanism may not be available. Yet when sufficient data from experimental observations can be obtained, even without knowing what exactly the physical meaning of the parameter settings or recorded variables are, data-driven methods can be used to construct minimal (and in a sense, robust) realizations of the system. The approach attempts, in a sense, to circumvent physical understanding, by building intrinsic “information geometries” of the observed data, and thus enabling prediction without physical/chemical knowledge. Here we use such an approach to obtain evolution equations for a data-driven realization of the original system – in effect, allowing prediction based on the informed interrogation of the agnostically organized observation database. We illustrate the approach on observations of (a) the normal form for the cusp singularity, (b) a cusp singularity for the nonisothermal CSTR, and (c) a random invertible transformation of the nonisothermal CSTR, showing that one can predict even when the observables are not “simply explainable” physical quantities. We discuss current limitations and possible extensions of the procedure.

show abstract

Reconstruction of normal forms by learning informed observation geometries from data

Cited by 68 publications

References 45 publications

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Synthesizing developmental trajectories

Data‐driven Evolution Equation Reconstruction for Parameter‐Dependent Nonlinear Dynamical Systems

Contact Info

Product

Resources

About