Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences

Characterizing the long term behavior of dynamical systems given limited measurements is a common challenge throughout the physical and biological sciences. This is a challenging task due to the sparsity and noise inherent to empirical observations, as well as the variability of possible long-term dynamics. We address this by introducing smooth prototype equivalences (SPE), a framework for matching sparse observations to prototypical behaviors using invertible neural networks which model smooth phase space deformations. SPE can localize the invariant sets describing long-term behavior of the observed dynamics through the learned mapping from prototype space to data space. Furthermore, SPE can classify dynamical regimes by comparing the data residual of the deformed measurements to prototype dynamics. Our method outperforms existing techniques in the classification of oscillatory systems and can efficiently identify invariant structures like limit cycles and fixed points in an equation-free manner, even when only a small, noisy subset of the phase space is observed. SPE further reveals driving genes in synthetic oscillators such as the repressilator regulatory circuit, and traces cyclic biological processes like the cell cycle trajectory directly from experimental high-dimensional single-cell gene expression data.