Multiple shooting for training neural differential equations on time series

TL;DR

This paper introduces a multiple shooting method to improve neural differential equation fitting for oscillatory time-series data, overcoming limitations of standard approaches that tend to flatten trajectories.

Contribution

The paper presents a novel multiple shooting approach for neural differential equations, enabling better fitting of oscillatory data where standard methods fail.

Findings

Multiple shooting improves fit quality for oscillatory data

Standard neural differential equations tend to produce flattened trajectories

Successful application to synthetic and experimental datasets

Abstract

Neural differential equations have recently emerged as a flexible data-driven/hybrid approach to model time-series data. This work experimentally demonstrates that if the data contains oscillations, then standard fitting of a neural differential equation may result in a flattened out trajectory that fails to describe the data. We then introduce the multiple shooting method and present successful demonstrations of this method for the fitting of a neural differential equation to two datasets (synthetic and experimental) that the standard approach fails to fit. Constraints introduced by multiple shooting can be satisfied using a penalty or augmented Lagrangian method.