Detecting the True Dimensionality of Complex Dynamical Systems Using Nonlinear Manifold Learning

Reduced macroscale modeling of complex dynamical systems is an integral part of many engineering disciplines. However, positing such reduced models is often difficult, as many systems are described using complex microscale models or via experimental data. Here, as a first step in writing accurate macroscale models for such systems, we use a data-driven methodology to extract the appropriate macroscopic variables with which to describe the dynamics. Because of the diversity and complexity of the potential systems of interest, we use nonlinear manifold learning techniques, which are flexible enough to accommodate a wide variety of structures within a data set, to extract these variables. However, traditional manifold learning techniques are plagued with the issue of "repeated eigendirections," where several of the computed modes parameterize the same direction in the data set. We propose an algorithm for automatically detecting these repeated eigendirections, allowing us to infer the true dimensionality of the data set and construct the most parsimonious reduced model of the underlying dynamical system. We demonstrate our approach using a model which describes cellular chemotaxis.