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.

Summary: The paper looks at the problem of modelling sequential data, specifically image data. It proposes to combine a (beta-)VAE model with a Neural ODE. The VAE encodes the input image to a location and velocity, the Neural ODE computes the dynamics over time, the VAE then decodes using the location parameters. To model the velocity, the authors extend the Neural ODE to be second order. The paper contains extensive introduction to the method, including ODE, VI, beta-VAE, generative models, ODE flow.

This paper combine several modeling ingredients (BNNs, ODEs, and VAEs) to produce a new family of models. It's not clear to my whether adding second-order dynamics in particular is advantageous over just adding extra latent dimensions to the state, which I think would be a generalization of the current approach. However, seeing a comparison against GPLVM-based models was nice, since these two approaches represent very different technical approaches to the same problem.

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.

Training dynamic models, such as neural ODEs, on long trajectories is a hard problem that requires using various tricks, such as trajectory splitting, to make model training work in practice. These methods are often heuristics with poor theoretical justifications, and require iterative manual tuning. We propose a principled multiple shooting technique for neural ODEs that splits the trajectories into manageable short segments, which are optimised in parallel, while ensuring probabilistic control on continuity over consecutive segments. We derive variational inference for our shooting-based latent neural ODE models and propose amortized encodings of irregularly sampled trajectories with a transformer-based recognition network with temporal attention and relative positional encoding. We demonstrate efficient and stable training, and state-of-the-art performance on multiple largescale benchmark datasets. Dynamical systems, from biological cells to weather, evolve according to their underlying mechanisms, often described by differential equations. In data-driven system identification we aim to learn the rules governing a dynamical system by observing the system for a time interval [0, T ], and fitting a model of the underlying dynamics to the observations by gradient descent. Such optimisation suffers from the curse of length: complexity of the loss function grows with the length of the observed trajectory (Ribeiro et al., 2020). For even moderate T the loss landscape can become highly complex and gradient descent fails to produce a good fit (Metz et al., 2021). To alleviate this problem previous works resort to cumbersome heuristics, such as iterative training and trajectory splitting (Yildiz et al., 2019; Kochkov et al., 2021; HAN et al., 2022; Lienen & Günnemann, 2022). The optimal control literature has a long history of multiple shooting methods, where the trajectory fitting is split into piecewise segments that are easy to optimise, with constraints to ensure continuity across the segments (van Domselaar & Hemker, 1975; Bock & Plitt, 1984; Baake et al., 1992).

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.