A statistical approach to latent dynamic modeling with differential
  equations
Maren Hackenberg Astrid Pechmann Clemens Kreutz Janbernd Kirschner Harald Binder
Ordinary differential equations (ODEs) can provide mechanistic models of
temporally local changes of processes, where parameters are often informed by
external knowledge. While ODEs are popular in systems modeling, they are less
established for statistical modeling of longitudinal cohort data, e.g., in a
clinical setting. Yet, modeling of local changes could also be attractive for
assessing the trajectory of an individual in a cohort in the immediate future
given its current status, where ODE parameters could be informed by further
characteristics of the individual. However, several hurdles so far limit such
use of ODEs, as compared to regression-based function fitting approaches. The
potentially higher level of noise in cohort data might be detrimental to ODEs,
as the shape of the ODE solution heavily depends on the initial value. In
addition, larger numbers of variables multiply such problems and might be
difficult to handle for ODEs. To address this, we propose to use each
observation in the course of time as the initial value to obtain multiple local
ODE solutions and build a combined estimator of the underlying dynamics. Neural
networks are used for obtaining a low-dimensional latent space for dynamic
modeling from a potentially large number of variables, and for obtaining
patient-specific ODE parameters from baseline variables. Simultaneous
identification of dynamic models and of a latent space is enabled by recently
developed differentiable programming techniques. We illustrate the proposed
approach in an application with spinal muscular atrophy patients and a
corresponding simulation study. In particular, modeling of local changes in
health status at any point in time is contrasted to the interpretation of
functions obtained from a global regression. This more generally highlights how
different application settings might demand different modeling strategies.