Deep dynamic modeling with just two time points: Can we still allow for individual trajectories?

TL;DR

This paper explores the potential of deep learning combined with differential equations to infer individual trajectories from extremely limited longitudinal biomedical data, specifically with only two time points per individual.

Contribution

It demonstrates how variational autoencoders linked to ODEs can recover individual trajectories and identify groups with similar patterns in small data settings.

Findings

Deep learning with ODEs can recover trajectories from two data points.

The approach can identify groups with similar individual trajectories.

Method performance depends on proper adaptation and regularity assumptions.

Abstract

Longitudinal biomedical data are often characterized by a sparse time grid and individual-specific development patterns. Specifically, in epidemiological cohort studies and clinical registries we are facing the question of what can be learned from the data in an early phase of the study, when only a baseline characterization and one follow-up measurement are available. Inspired by recent advances that allow to combine deep learning with dynamic modeling, we investigate whether such approaches can be useful for uncovering complex structure, in particular for an extreme small data setting with only two observations time points for each individual. Irregular spacing in time could then be used to gain more information on individual dynamics by leveraging similarity of individuals. We provide a brief overview of how variational autoencoders (VAEs), as a deep learning approach, can be linked to ordinary differential equations (ODEs) for dynamic modeling, and then specifically investigate the feasibility of such an approach that infers individual-specific latent trajectories by including regularity assumptions and individuals' similarity. We also provide a description of this deep learning approach as a filtering task to give a statistical perspective. Using simulated data, we show to what extent the approach can recover individual trajectories from ODE systems with two and four unknown parameters and infer groups of individuals with similar trajectories, and where it breaks down. The results show that such dynamic deep learning approaches can be useful even in extreme small data settings, but need to be carefully adapted.