Positive Latent Suppression in Logistic Neural Differential Equations: Trajectory Reconstruction, Identifiability and Population Hindcasts

A close fit to a scalar time series does not establish either accurate extrapolation or unique mechanistic parameter recovery. We examine a logistic gray-box differential equation coupled to a positive latent suppression state whose relative rate is represented by a bounded neural network. Positivity at every finite time, a population upper bound and a local turning-point criterion are established. For freely chosen bounded closures on a finite observation interval, compatible mechanistic parameters are characterized by a convex feasible set and their nonuniqueness is established. With a positive closure bound, unrestricted positive growth parameters and a free latent initial value, every positive twice continuously differentiable trajectory on that interval is compatible with such a closure. Strictly feasible parameter pairs have two-dimensional feasible neighborhoods. These results describe identifiability boundaries in the freely chosen closure class; a complete structural-identifiability analysis of the fixed finite-width neural architecture remains a separate problem, as does selection of a unique future continuation. Numerical evaluation includes four synthetic mechanisms and current-vintage population estimates for China, India and Japan. Across six historical evaluation windows per country, the gray-box model attains mean training and test absolute percentage errors of 1.126% and 1.338%, respectively. Its mean test error in these windows is below five of the six comparison implementations but above the 0.684% of a residual universal differential equation. Equal additional training of these two models reduces their test errors to 0.714% and 0.409%. The results support conditional usefulness of the latent-suppression structure while demonstrating limitations in optimization, parameter interpretation and temporal generalization.