Positive Latent Suppression in Logistic Neural Differential Equations: Trajectory Reconstruction, Identifiability and Population Hindcasts
- 1School of Science, Inner Mongolia University of Technology, Hohhot 010051, China
- 2Department of Mathematics, Shanghai University, Shanghai 200444, China
Abstract
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.
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BibTeX
@article{xu2026positive,
title = {{Positive Latent Suppression in Logistic Neural Differential Equations: Trajectory Reconstruction, Identifiability and Population Hindcasts}},
author = {Chenhao Xu and Tiecheng Xia and Haijing Ma and Junjie Fan},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213898},
url = {https://www.mdpi.com/2227-7390/14/21/3898},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Positive Latent Suppression in Logistic Neural Differential Equations: Trajectory Reconstruction, Identifiability and Population Hindcasts
AU - Chenhao Xu
AU - Tiecheng Xia
AU - Haijing Ma
AU - Junjie Fan
PY - 2026
DA - 2026/10/26
JO - Mathematics
DO - 10.3390/math14213898
UR - https://www.mdpi.com/2227-7390/14/21/3898
AB - 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.
SN - 2227-7390
ER -