Homotopy-based training of NeuralODEs for accurate dynamics discovery

Joon-Hyuk Ko, Hankyul Koh, Nojun Park, Wonho Jhe

Keywords: neural ordinary differential equation, synchronization, homotopy, dynamical systems, physics

TL;DR: Building upon ideas from the chaos literature, we introduce a novel method of training neural ordinary differential equations with drastic improvements for long complex time series prediction.

Abstract: Conceptually, Neural Ordinary Differential Equations (NeuralODEs) pose an attractive way to extract dynamical laws from time series data, as they are natural extensions of the traditional differential equation-based modeling paradigm of the physical sciences. In practice, NeuralODEs display long training times and suboptimal results, especially for longer duration data where they may fail to fit the data altogether. While methods have been proposed to stabilize NeuralODE training, many of these involve placing a strong constraint on the functional form the trained NeuralODE can take that the actual underlying governing equation does not guarantee satisfaction. In this work, we present a novel NeuralODE training algorithm that leverages tools from the chaos and mathematical optimization communities -- synchronization and homotopy optimization -- for a breakthrough in tackling the NeuralODE training obstacle. We demonstrate architectural changes are unnecessary for effective NeuralODE training. Compared to the conventional training methods, our algorithm achieves drastically lower loss values without any changes to the model architectures. Experiments on both simulated and real systems with complex temporal behaviors demonstrate NeuralODEs trained with our algorithm are able to accurately capture true long term behaviors and correctly extrapolate into the future.

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