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Data-driven discovery of intrinsic dynamics

A preprint version of the article is available at arXiv.


Dynamical models underpin our ability to understand and predict the behaviour of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and, in data-driven cases, the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behaviour of the system. In truth these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system’s intrinsic state variables directly from time-series data, as well as predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together—a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behaviour. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system.

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Fig. 1: A representation of a manifold.
Fig. 2: The three basic steps of our method to learn an atlas of charts and dynamics.
Fig. 3: Quasiperiodic dynamics on the surface of a torus.
Fig. 4: A spiral wave from a lambda-omega reaction–diffusion system.
Fig. 5: Bursting data from the K−S system.

Data availability

Data are available at (ref. 63). For data that are unavailable due to size restrictions, code that exactly reproduces the data has been deposited in the same repository.

Code availability

Code is available at (ref. 63).


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We acknowledge the use of the Sabine cluster from the Research Computing Data Core at the University of Houston, and the assistance of D. A. Kaji and the Luke cluster. This work was supported by the Air Force Office of Scientific Research (grant no. FA9550-18-0174 to M.D.G.) and an Office of Naval Research grant (grant no. N00014-18-1-2865, Vannevar Bush Faculty Fellowship, to M.D.G.).

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D.F. and M.D.G. designed and performed the research, analysed the data and wrote the paper.

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Correspondence to Daniel Floryan or Michael D. Graham.

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Nature Machine Intelligence thanks Stefan Schonscheck and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Video 1

From left to right: simulation of equation (5) in the main text; model prediction using method of ref. 46, with two degrees of freedom; model prediction using method from ref. 46, with one degree of freedom; model prediction with the current approach and one degree of freedom with three charts.

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Floryan, D., Graham, M.D. Data-driven discovery of intrinsic dynamics. Nat Mach Intell 4, 1113–1120 (2022).

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