Dynamics, data and reconstruction

Data-driven learning is prevalent in many fields of science, mathematics and engineering. The goal of data-driven learning of dynamical systems is to interpret timeseries as a continuous observation of an underlying dynamical system. This task is not well-posed for a variety of reasons. A dynamical system may have multiple sub-systems co-existing within it. The nature of the dataset depends on the portion of the phase space being viewed, and may thus be confined to a sub-system. Secondly these sub-systems may be topologically inter-weaved, so may be inseparable computationally. Thirdly, two timeseries sampled separately from different dynamical systems may be close or even indistinguishable. Thus a timeseries may not have a unique source. We show how these ambiguities are circumvented if one considers dynamical systems and measurement maps collectively. This is made possible in a category theoretical framework, in which reconstruction is unique up to equivalences. Dynamical systems, observed dynamical systems, and timeseries data - each of these three collections have an extensive network of relations within them, which gives them the mathematical structure of a category. We show that the transformations preserve these relations, and thus have the mathematical property of being functors. Secondly we show that under mild conditions of consistency, reconstruction algorithms are themselves functors from the category of timeseries-data into the category of dynamical systems. Finally we present the task of inverting the data into dynamics, using the language of Kan extensions.