Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory

Wulkow, Niklas; Koltai, Péter; Sunkara, Vikram; Schütte, Christof

doi:10.48550/arxiv.2112.06742

Cited by 2 publications

(4 citation statements)

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“…In addition, it should be worthwhile to use the combined solution of the (SPA 1) and (SPA 2) problems as done in [23] and observe whether this improves the performance. Furthermore, constructing a nonlinear SPA model consisting of the multiplication of a linear mapping with a nonlinear function, as done in [26], could give an improved model accuracy and, therefore, a more reliable quantification of influences. Lastly, since in this article, variables were always expressed using a higher number of landmark points, it should be interesting to investigate whether for high-dimensional variables, projecting them to a low-dimensional representation using SPA and performing the same dependency analysis is still sensible.…”

Section: Discussionmentioning

confidence: 99%

“…It was discussed in [26] that for K ≤ D, the representation of points in this way is the orthogonal projection onto a convex polytope with vertices given by the columns of Σ. The coordinates γ then specify the position of this projection with respect to the vertices of the polytope and are called Barycentric Coordinates (BCs).…”

Section: Representation Of the Data In Barycentric Coordinatesmentioning

confidence: 99%

“…For K > D + 1 and given landmark points, the representation with barycentric coordinates is generally not unique (while the set of points that can be represented by Σγ is a convex polytope, some landmark points can even lie inside this polytope if K > D + 1). Therefore, let us define the representation of a point X analogously to [26] in the following way:…”

Section: Remarkmentioning

confidence: 99%

See 2 more Smart Citations

Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

Wulkow

2022

AppliedMath

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A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points using the Scalable Probabilistic Approximation algorithm. This is followed by the construction of a linear mapping between these affiliations for different variables and forward in time. This linear mapping, or matrix, can be directly interpreted in light of unidirectional dependencies, and relevant properties of it are quantified. These quantifications, given by the sum of singular values and the average row variance of the matrix, then serve as measures for the influences between variables of the dynamics. The validity of the method is demonstrated with theoretical results and on several numerical examples, covering deterministic, stochastic, and delayed types of dynamics. Moreover, the method is applied to a non-classical example given by real-world basketball player movement, which exhibits highly random movement and comes without a physical intuition, contrary to many examples from, e.g., life sciences.

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Section: Discussionmentioning

confidence: 99%

Section: Representation Of the Data In Barycentric Coordinatesmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

Wulkow

2022

AppliedMath

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“…There are also several other relevant papers including [55], which uses a recurrent neural network to predict chaotic systems including Lorenz 63. In [59], the authors discuss memory length and Takens' theorem [50] in the context of learning Lorenz 96. In addition, [48] considers approximating chaotic dynamical systems using limited data and external forcing.…”

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confidence: 99%

Deep Learning of Chaotic Systems from Partially-Observed Data

Churchill¹,

Xiu²

2022

Preprint

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Recently, a general data driven numerical framework has been developed for learning and modeling of unknown dynamical systems using fully-or partially-observed data. The method utilizes deep neural networks (DNNs) to construct a model for the flow map of the unknown system. Once an accurate DNN approximation of the flow map is constructed, it can be recursively executed to serve as an effective predictive model of the unknown system. In this paper, we apply this framework to chaotic systems, in particular the well-known Lorenz 63 and 96 systems, and critically examine the predictive performance of the approach. A distinct feature of chaotic systems is that even the smallest perturbations will lead to large (albeit bounded) deviations in the solution trajectories. This makes long-term predictions of the method, or any data driven methods, questionable, as the local model accuracy will eventually degrade and lead to large pointwise errors. Here we employ several other qualitative and quantitative measures to determine whether the chaotic dynamics have been learned. These include phase plots, histograms, autocorrelation, correlation dimension, approximate entropy, and Lyapunov exponent. Using these measures, we demonstrate that the flow map based DNN learning method is capable of accurately modeling chaotic systems, even when only a subset of the state variables are available to the DNNs. For example, for the Lorenz 96 system with 40 state variables, when data of only 3 variables are available, the method is able to learn an effective DNN model for the 3 variables and produce accurately the chaotic behavior of the system.

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Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory

Cited by 2 publications

References 29 publications

Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations

Deep Learning of Chaotic Systems from Partially-Observed Data

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