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）已被证明可用于学习任意轨迹的非线性动力学。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。本文提出了一种名为调制神经常微分方程（MoNODEs）的新框架，该框架将动力学状态与潜在的静态变化因子区分开来，并改进了现有NODE方法。具体而言，我们引入了从数据中学习的$\textit{时不变调制变量}$。我们将所提出的框架整合到四种现有的NODE变体中。我们在振荡系统、视频及人类行走轨迹（每个轨迹具有特定的轨迹调制）上测试了MoNODE。我们的框架持续提升了现有模型泛化到新动态参数化以及进行远时域预测的能力。此外，我们验证了所提出的调制变量能够通过$R^2$得分有效反映真实未知变化因子的信息。