# Modulated Neural ODEs

## TL;DR

Modulated Neural ODEs introduce learned, time-invariant modulator variables to enhance the modeling of trajectory variations and improve generalization and forecasting in dynamic systems.

## Contribution

The paper proposes a novel framework, MoNODEs, that separates static factors from dynamics in NODEs using learned modulator variables, enhancing existing models.

## Key findings

- MoNODEs improve generalization to new dynamic parameters.
- MoNODEs enhance far-horizon forecasting accuracy.
- Modulator variables are informative of true factors of variation.

## Abstract

Neural ordinary differential equations (NODEs) have been proven useful for learning non-linear dynamics of arbitrary trajectories. However, current NODE methods capture variations across trajectories only via the initial state value or by auto-regressive encoder updates. In this work, we introduce Modulated Neural ODEs (MoNODEs), a novel framework that sets apart dynamics states from underlying static factors of variation and improves the existing NODE methods. In particular, we introduce $\textit{time-invariant modulator variables}$ that are learned from the data. We incorporate our proposed framework into four existing NODE variants. We test MoNODE on oscillating systems, videos and human walking trajectories, where each trajectory has trajectory-specific modulation. Our framework consistently improves the existing model ability to generalize to new dynamic parameterizations and to perform far-horizon forecasting. In addition, we verify that the proposed modulator variables are informative of the true unknown factors of variation as measured by $R^2$ scores.