# ODE2VAE: Deep generative second order ODEs with Bayesian neural networks

arXiv: https://arxiv.org/abs/1905.10994 · HTML: https://mlsanity.com/paper/1905.10994/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2019.

## Built on by (most cited first, 30 of 36)

- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Continuous-Time Model-Based Reinforcement Learning](https://mlsanity.com/paper/2102.04764/) (2021)
- [LEADS: Learning Dynamical Systems that Generalize Across Environments](https://mlsanity.com/paper/2106.04546/) (2021)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Latent SDEs on Homogeneous Spaces](https://mlsanity.com/paper/2306.16248/) (2023)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [Contrastive Conditional Neural Processes](https://mlsanity.com/paper/2203.03978/) (2022)
- [Non-equispaced Fourier Neural Solvers for PDEs](https://mlsanity.com/paper/2212.04689/) (2022)
- [Continuous-Depth Neural Models for Dynamic Graph Prediction](https://mlsanity.com/paper/2106.11581/) (2021)
- [Controllable Video Generation by Learning the Underlying Dynamical System with Neural ODE](https://mlsanity.com/paper/2303.05323/) (2023)
- [Segmenting Hybrid Trajectories using Latent ODEs](https://mlsanity.com/paper/2105.03835/) (2021)
- [VidStyleODE: Disentangled Video Editing via StyleGAN and NeuralODEs](https://mlsanity.com/paper/2304.06020/) (2023)
- [TaylorSwiftNet: Taylor Driven Temporal Modeling for Swift Future Frame Prediction](https://mlsanity.com/paper/2110.14392/) (2021)
- [Variational Mixtures of ODEs for Inferring Cellular Gene Expression Dynamics](https://mlsanity.com/paper/2207.04166/) (2022)
- [Meta-learning using privileged information for dynamics](https://mlsanity.com/paper/2104.14290/) (2021)
- [STR-GODEs: Spatial-Temporal-Ridership Graph ODEs for Metro Ridership Prediction](https://mlsanity.com/paper/2107.04980/) (2021)
- [Continuous-Time Video Generation via Learning Motion Dynamics with Neural ODE](https://mlsanity.com/paper/2112.10960/) (2021)
- [Imbedding Deep Neural Networks](https://mlsanity.com/paper/2202.00113/) (2022)
- [Continual Learning of Multi-modal Dynamics with External Memory](https://mlsanity.com/paper/2203.00936/) (2022)
- [STONet: A Neural-Operator-Driven Spatio-temporal Network](https://mlsanity.com/paper/2204.08414/) (2022)
- [Hawkes Process Based on Controlled Differential Equations](https://mlsanity.com/paper/2305.07031/) (2023)
- [Learning Latent Dynamics via Invariant Decomposition and (Spatio-)Temporal Transformers](https://mlsanity.com/paper/2306.12077/) (2023)
- [Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning](https://mlsanity.com/paper/2405.09453/) (2024)
- [Particle Filter Bridge Interpolation](https://mlsanity.com/paper/2103.14963/) (2021)

## Abstract (arXiv)

We present Ordinary Differential Equation Variational Auto-Encoder (ODE2VAE), a latent second order ODE model for high-dimensional sequential data. Leveraging the advances in deep generative models, ODE2VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex continuous-time latent dynamics. Our model explicitly decomposes the latent space into momentum and position components and solves a second order ODE system, which is in contrast to recurrent neural network (RNN) based time series models and recently proposed black-box ODE techniques. In order to account for uncertainty, we propose probabilistic latent ODE dynamics parameterized by deep Bayesian neural networks. We demonstrate our approach on motion capture, image rotation and bouncing balls datasets. We achieve state-of-the-art performance in long term motion prediction and imputation tasks.