(61e) Nonlinear System Identification: Finding Normal Forms By Iteratively Uncovering Informed Geometries

We present a data-driven technique with roots in manifold learning for recovering the underlying structure of nonlinear dynamical systems. The data from such a dynamical system can be organized along three axes: the many trials with different parameter settings, the many measurement channels associated with each trial, and the many time instances. We assume no knowledge of the dimensionality or organization of the parameter settings or the measurement channels; we only know an index identifying the parameter trial, an index identifying the measurement channel, and an index describing the measurement time.

Our algorithm analyzes the structure of the data along any two of the axes and uses the results to construct a more informed geometry for the third axis. This process can proceed iteratively, improving the geometry for all three axes. We demonstrate a version using multilevel clustering to structure the data, as well as a version using diffusion maps. Using an earth mover's distance in another version of the algorithm can even overcome scenarios where the channel indices for the different trials are scrambled. We also show that applying our algorithm to invertible functions of the data gives rise to homeomorphic (or even isometric) embeddings conveying the same prototypical behavior.