Experimental measurements of physical systems often have a finite number of independent channels, causing essential dynamical variables to remain unobserved. However, many popular methods for unsupervised inference of latent dynamics from experimental data implicitly assume that the measurements have higher intrinsic dimensionality than the underlying system---making coordinate identification a dimensionality reduction problem. Here, we study the opposite limit, in which hidden governing coordinates must be inferred from only a low-dimensional time series of measurements. Inspired by classical techniques for studying the strange attractors of chaotic systems, we introduce a general embedding technique for time series, consisting of an autoencoder trained with a novel latent-space loss function. We first apply our technique to a variety of synthetic and real-world datasets with known strange attractors, and we use established and novel measures of attractor fidelity to show that our method successfully reconstructs attractors better than existing techniques . We then use our technique to discover dynamical attractors in datasets ranging from patient electrocardiograms, to household electricity usage, to eruptions of the Old Faithful geyser---demonstrating diverse applications of our technique for exploratory data analysis.
Experimental measurements of physical systems often have a limited number of independent channels, causing essential dynamical variables to remain unobserved. However, many popular methods for unsupervised inference of latent dynamics from experimental data implicitly assume that the measurements have higher intrinsic dimensionality than the underlying system---making coordinate identification a dimensionality reduction problem. Here, we study the opposite limit, in which hidden governing coordinates must be inferred from only a low-dimensional time series of measurements. Inspired by classical techniques for studying the strange attractors of chaotic systems, we introduce a general embedding technique for time series, consisting of an autoencoder trained with a novel latent-space loss function. We show that our technique reconstructs the strange attractors of synthetic and real-world systems better than existing techniques , and that it creates consistent, predictive representations of even stochastic systems. We conclude by using our technique to discover dynamical attractors in diverse systems such as patient electrocardiograms, household electricity usage, and eruptions of the Old Faithful geyser---demonstrating diverse applications of our technique for exploratory data analysis.
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