The Empirical Bayes Variational Autoencoder—A Neural <scp>ODE</scp> Approach for Population Modeling in Pharmacology

Variational autoencoders (VAEs) combined with neural ordinary differential equations provide a flexible framework for exploring neural latent-variable models in population pharmacokinetics. In this work, we investigate an empirical Bayes VAE formulation that integrates encoder-decoder architectures with covariate-dependent population priors, enabling correlated latent representations and probabilistic inference. We evaluate the proposed framework using controlled simulation studies and a small clinical benchmark dataset. The simulation experiments assess the ability to recover known population structures and covariate effects, while the clinical study evaluates subject-specific prediction and model diagnostics. In simulation studies with correlated individual parameters, the empirical Bayes VAE consistently captured population-level variability, whereas a fixed-prior VAE baseline exhibited systematic biases. In our experiments, extrapolation beyond the training dosing schedules showed more stable predictive behavior when using the proposed input-response normalization, relative to models trained without normalization, within a limited range. Diagnostic analyses indicated clear relationships between inferred latent variables and true parameters, and estimated observation noise was consistent with simulated values. In the clinical case study, cross-validation experiments suggested predictive performance comparable to previously reported neural ODE-based approaches. Overall, the results illustrate the feasibility of combining empirical Bayes inference with neural ODE-based decoders for population modeling. The proposed framework should be viewed as a methodological proof-of-concept, highlighting both the potential and the current limitations of variational neural approaches in pharmacometric applications.