A non-autonomous equation discovery method for time signal classification

TL;DR

This paper introduces a novel framework for time signal classification using non-autonomous dynamical equations, leveraging function dictionaries and the adjoint method for efficient training and stability analysis, with demonstrated effectiveness on synthetic and real data.

Contribution

The paper presents a new non-autonomous dynamical system approach for time signal classification, including stability analysis and interpretability, with significantly fewer parameters than existing methods.

Findings

Achieves comparable accuracy with fewer parameters.

Fourier dictionaries outperform polynomial ones.

Provides graphical interpretability via phase portraits.

Abstract

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.