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Ordinary differential equations (ODEs) can provide mechanistic models oftemporally local changes of processes, where parameters are often informed byexternal knowledge. While ODEs are popular in systems modeling, they are lessestablished for statistical modeling of longitudinal cohort data, e.g., in aclinical setting. Yet, modeling of local changes could also be attractive forassessing the trajectory of an individual in a cohort in the immediate futuregiven its current status, where ODE parameters could be informed by furthercharacteristics of the individual. However, several hurdles so far limit suchuse of ODEs, as compared to regression-based function fitting approaches. Thepotentially 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. Inaddition, larger numbers of variables multiply such problems and might bedifficult to handle for ODEs. To address this, we propose to use eachobservation in the course of time as the initial value to obtain multiple localODE solutions and build a combined estimator of the underlying dynamics. Neuralnetworks are used for obtaining a low-dimensional latent space for dynamicmodeling from a potentially large number of variables, and for obtainingpatient-specific ODE parameters from baseline variables. Simultaneousidentification of dynamic models and of a latent space is enabled by recentlydeveloped differentiable programming techniques. We illustrate the proposedapproach in an application with spinal muscular atrophy patients and acorresponding simulation study. In particular, modeling of local changes inhealth status at any point in time is contrasted to the interpretation offunctions obtained from a global regression. This more generally highlights howdifferent application settings might demand different modeling strategies.

A statistical approach to latent dynamic modeling with differential equations