Identifying Latent Stochastic Differential Equations

TL;DR

This paper introduces a self-supervised variational autoencoder framework to identify and recover latent stochastic differential equations from high-dimensional time series data, enabling understanding of underlying dynamics.

Contribution

It proposes a novel method combining variational autoencoders and SDE theory to recover latent variables and SDE coefficients, with proven identifiability in the infinite data limit.

Findings

Successfully recovers underlying SDEs in simulated video tasks

Demonstrates applicability to real-world datasets

Achieves theoretical identifiability of latent variables

Abstract

We present a method for learning latent stochastic differential equations (SDEs) from high-dimensional time series data. Given a high-dimensional time series generated from a lower dimensional latent unknown It\^o process, the proposed method learns the mapping from ambient to latent space, and the underlying SDE coefficients, through a self-supervised learning approach. Using the framework of variational autoencoders, we consider a conditional generative model for the data based on the Euler-Maruyama approximation of SDE solutions. Furthermore, we use recent results on identifiability of latent variable models to show that the proposed model can recover not only the underlying SDE coefficients, but also the original latent variables, up to an isometry, in the limit of infinite data. We validate the method through several simulated video processing tasks, where the underlying SDE is known, and through real world datasets.