Data collection is often constrained by budget, time, and participant burden, resulting in sparse longitudinal measurements. These constraints call for designs that collect observations at the most informative time points to improve inference and downstream decisions. However, most existing designs rely heavily on plug-in estimates and provide limited mechanisms to account for uncertainty in parameter estimation. To tackle this challenge, this study proposes a multi-layer Gaussian process framework that incorporates uncertainty directly into the design process and captures crosssubject dependence and individual-level variability. This framework further derives a closed-form utility from Shannon information gain and provides a principled and lowcost criterion for the sequential selection of observation times, opening avenues for integrating learning into AI-enabled decision systems.