  In several crucial applications, domain knowledge is encoded by a system of
ordinary differential equations (ODE), often stemming from underlying physical
and biological processes. A motivating example is intensive care unit patients:
the dynamics of vital physiological functions, such as the cardiovascular
system with its associated variables (heart rate, cardiac contractility and
output and vascular resistance) can be approximately described by a known
system of ODEs. Typically, some of the ODE variables are directly observed
(heart rate and blood pressure for example) while some are unobserved (cardiac
contractility, output and vascular resistance), and in addition many other
variables are observed but not modeled by the ODE, for example body
temperature. Importantly, the unobserved ODE variables are known-unknowns: We
know they exist and their functional dynamics, but cannot measure them
directly, nor do we know the function tying them to all observed measurements.
As is often the case in medicine, and specifically the cardiovascular system,
estimating these known-unknowns is highly valuable and they serve as targets
for therapeutic manipulations. Under this scenario we wish to learn the
parameters of the ODE generating each observed time-series, and extrapolate the
future of the ODE variables and the observations. We address this task with a
variational autoencoder incorporating the known ODE function, called GOKU-net
for Generative ODE modeling with Known Unknowns. We first validate our method
on videos of single and double pendulums with unknown length or mass; we then
apply it to a model of the cardiovascular system. We show that modeling the
known-unknowns allows us to successfully discover clinically meaningful
unobserved system parameters, leads to much better extrapolation, and enables
learning using much smaller training sets.
