[Paper Review] ODE2VAE: Deep generative second order ODEs with Bayesian neural networks

Çağatay Yıldız, Markus Heinonen+1 more|arXiv (Cornell University)|Jan 1, 2019

Generative Adversarial Networks and Image Synthesis14 references60 citations

TL;DR

ODE2VAE introduces a deep generative model that uses second-order ordinary differential equations (ODEs) with Bayesian neural networks to learn continuous-time latent dynamics from high-dimensional sequential data. By explicitly modeling momentum and position in a probabilistic latent ODE framework, it achieves state-of-the-art performance in long-term motion prediction and data imputation on motion capture, image rotation, and bouncing balls datasets.

ABSTRACT

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.

Motivation & Objective

• To model complex, continuous-time dynamics in high-dimensional sequential data using a structured latent space.
• To improve long-term motion prediction and data imputation by explicitly modeling momentum and position in a second-order ODE framework.
• To incorporate uncertainty quantification in latent dynamics using deep Bayesian neural networks.
• To overcome limitations of RNNs and black-box ODE approaches by enforcing physical plausibility through second-order dynamics.
• To demonstrate state-of-the-art performance on benchmark sequential data tasks such as motion capture and image rotation.

Proposed method

• The model decomposes the latent space into momentum and position components, enabling a second-order ODE system to describe continuous-time dynamics.
• It parameterizes the ODE dynamics using deep Bayesian neural networks to model uncertainty in the learned dynamics.
• The latent ODE system is trained end-to-end via a variational auto-encoder (VAE) objective, optimizing both reconstruction and prior regularization.
• The model uses stochastic gradient variational Bayes (SGVB) for approximate posterior inference, enabling scalable training on high-dimensional sequences.
• The second-order ODE formulation allows for more expressive and physically interpretable dynamics compared to first-order ODEs or RNNs.
• The framework supports both generation and imputation of sequential data by solving the learned ODEs forward or backward in time.

Experimental results

Research questions

• RQ1 Can a second-order ODE-based latent model outperform first-order ODEs and RNNs in modeling complex sequential dynamics?
• RQ2 How well can a Bayesian neural network parameterization of latent ODEs handle uncertainty in high-dimensional time series?
• RQ3 Can explicit momentum and position decomposition improve long-term motion prediction and data imputation?
• RQ4 To what extent does the second-order ODE structure enhance the interpretability and generalization of learned dynamics?
• RQ5 Does the proposed ODE2VAE framework achieve state-of-the-art performance on benchmark sequential data tasks?

Key findings

• ODE2VAE achieves state-of-the-art performance in long-term motion prediction on motion capture and image rotation datasets.
• The model demonstrates superior data imputation capabilities, effectively recovering missing segments in sequential trajectories.
• The use of second-order ODEs with Bayesian neural networks enables accurate uncertainty quantification in latent dynamics.
• Explicit momentum and position decomposition leads to more stable and physically plausible latent trajectories compared to first-order models.
• The framework generalizes well across diverse sequential data types, including motion capture, image rotation, and bouncing balls.
• The integration of deep Bayesian networks into the ODE2VAE framework improves robustness to noisy or incomplete observations.