Identifying Latent Stochastic Differential Equations with Variational Auto-Encoders

Variational auto-encoders (VAEs) are a widely used tool to learn lower-dimensional latent representations of high-dimensional data. However, the learned latent representations often lack interpretability, and it is challenging to extract relevant information from the representation of the dataset in the latent space. In particular, when the high-dimensional data is governed by unknown and lower-dimensional dynamics, arising, for instance, from unknown physical or biological interactions, the latent space representation often fails to bring insight on these dynamics. We propose a VAE-based framework for recovering latent dynamics governed by stochastic differential equations (SDEs). Our motivation for using SDEs is that they are already often used to model physical and biological phenomena, to study financial markets, and their properties have been extensively studied in the fields of probability and statistics. We believe this method can be useful in describing trajectories of high dimensional data with underlying physical or biological dynamics, with applications such as video data, longitudinal medical data or gene regulatory dynamics.