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.

翻译：神经常微分方程（NODEs）已被证明对于学习任意轨迹的非线性动力学具有重要作用。然而，当前NODE方法仅通过初始状态值或自回归编码器更新来捕捉轨迹间的变化。本文提出调制式神经常微分方程（MoNODEs），这是一个将动力学状态与潜在静态变化因素相区分的新框架，并对现有NODE方法进行了改进。具体而言，我们引入了从数据中学习的$\textit{时不变调节变量}$。我们将所提出的框架融入四种现有的NODE变体中，并在振荡系统、视频及人类行走轨迹上测试了MoNODE，其中每条轨迹都具有轨迹特定的调制。我们的框架持续提升了现有模型泛化至新动态参数化以及进行远期预测的能力。此外，我们验证了所提出的调节变量能够有效反映真实的未知变化因素，其信息含量由$R^2$分数衡量。