A Statistical Approach to Latent Dynamic Modeling with Differential Equations

Ordinary differential equations (ODEs) provide mechanistic models of temporally local changes of processes, where parameters can be informed by external knowledge. For statistical modeling of longitudinal cohort data, the use of ODEs is so far limited compared to regression-based global function fitting approaches, yet modeling local changes based on an individual’s current status with ODEs could also be attractive in a clinical cohort setting. This is potentially due to a larger number of variables to be modeled and a higher noise level, as the shape of an ODE solution strongly depends on the initial value. To address this, we propose to use each observation as the initial value to obtain multiple local ODE solutions, supporting subsequent prediction starting from arbitrary time points, and build a combined estimator. We use neural networks for obtaining a low-dimensional latent space for dynamic modeling with many variables, and for obtaining individual-specific ODE parameters from baseline variables. Differentiable programming allows for simultaneously fitting dynamic models and a latent space. We illustrate the approach in an application with spinal muscular atrophy patients and contrast modeling of local changes in health status to global fitting via regression, highlighting how different application settings might demand different modeling strategies.