  Certain neural network architectures, in the infinite-layer limit, lead to
systems of nonlinear differential equations. Motivated by this idea, we develop
a framework for analyzing time signals based on non-autonomous dynamical
equations. We view the time signal as a forcing function for a dynamical system
that governs a time-evolving hidden variable. As in equation discovery, the
dynamical system is represented using a dictionary of functions and the
coefficients are learned from data. This framework is applied to the time
signal classification problem. We show how gradients can be efficiently
computed using the adjoint method, and we apply methods from dynamical systems
to establish stability of the classifier. Through a variety of experiments, on
both synthetic and real datasets, we show that the proposed method uses orders
of magnitude fewer parameters than competing methods, while achieving
comparable accuracy. We created the synthetic datasets using dynamical systems
of increasing complexity; though the ground truth vector fields are often
polynomials, we find consistently that a Fourier dictionary yields the best
results. We also demonstrate how the proposed method yields graphical
interpretability in the form of phase portraits.
