# Modulated Neural ODEs

arXiv: https://arxiv.org/abs/2302.13262 · HTML: https://mlsanity.com/paper/2302.13262/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 38 references that are in the map.
- First posted on arXiv 26 Feb 2023.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Stochastic Latent Residual Video Prediction](https://mlsanity.com/paper/2002.09219/) (2020)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [SIMONe: View-Invariant, Temporally-Abstracted Object Representations via Unsupervised Video Decomposition](https://mlsanity.com/paper/2106.03849/) (2021)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)
- [Visual Representation Learning Does Not Generalize Strongly Within the Same Domain](https://mlsanity.com/paper/2107.08221/) (2021)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Learning Invariances using the Marginal Likelihood](https://mlsanity.com/paper/1808.05563/) (2018)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Vid-ODE: Continuous-Time Video Generation with Neural Ordinary Differential Equation](https://mlsanity.com/paper/2010.08188/) (2020)
- [Last Layer Marginal Likelihood for Invariance Learning](https://mlsanity.com/paper/2106.07512/) (2021)
- [Latent Neural ODEs with Sparse Bayesian Multiple Shooting](https://mlsanity.com/paper/2210.03466/) (2022)
- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [Deep Temporal Sigmoid Belief Networks for Sequence Modeling](https://mlsanity.com/paper/1509.07087/) (2015)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [Simple Video Generation using Neural ODEs](https://mlsanity.com/paper/2109.03292/) (2021)

## Built on by (most cited first, 10 of 10)

- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [Discovering Physics Laws of Dynamical Systems via Invariant Function Learning](https://mlsanity.com/paper/2502.04495/) (2025)
- [Latent Space Energy-based Neural ODEs](https://mlsanity.com/paper/2409.03845/) (2024)
- [Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations](https://mlsanity.com/paper/2410.10174/) (2024)
- [System-Aware Neural ODE Processes for Few-Shot Bayesian Optimization](https://mlsanity.com/paper/2406.02352/) (2024)
- [DyMoDreamer: World Modeling with Dynamic Modulation](https://mlsanity.com/paper/2509.24804/) (2025)
- [Optimization as a Dynamical System: Generative Schedules from Latent ODEs](https://mlsanity.com/paper/2509.23052/) (2025)
- [ODeform: Learning Continuous 4D Motion for Shape Deformation with Neural ODEs](https://mlsanity.com/paper/2607.20670/) (2026)
- [Flowing Through States: Neural ODE Regularization for Reinforcement Learning](https://mlsanity.com/paper/2608.06595/) (2026)
- [Identifying ODEs from Unstructured Data with Causal Representation Learning](https://mlsanity.com/paper/2609.37083/) (2026)

## Abstract (arXiv)

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 textittime-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 R2 scores.