  End-to-end learning of dynamical systems with black-box models, such as
neural ordinary differential equations (ODEs), provides a flexible framework
for learning dynamics from data without prescribing a mathematical model for
the dynamics. Unfortunately, this flexibility comes at the cost of
understanding the dynamical system, for which ODEs are used ubiquitously.
Further, experimental data are collected under various conditions (inputs),
such as treatments, or grouped in some way, such as part of sub-populations.
Understanding the effects of these system inputs on system outputs is crucial
to have any meaningful model of a dynamical system. To that end, we propose a
structured latent ODE model that explicitly captures system input variations
within its latent representation. Building on a static latent variable
specification, our model learns (independent) stochastic factors of variation
for each input to the system, thus separating the effects of the system inputs
in the latent space. This approach provides actionable modeling through the
controlled generation of time-series data for novel input combinations (or
perturbations). Additionally, we propose a flexible approach for quantifying
uncertainties, leveraging a quantile regression formulation. Results on
challenging biological datasets show consistent improvements over competitive
baselines in the controlled generation of observational data and inference of
biologically meaningful system inputs.
