Hybrid Neural Differential Equations to Model Unknown Mechanisms and States in Biology

Abstract

Hybrid neural differential equations (HNDEs) embed neural network components within mechanistic scaffolds, combining the structural interpretability of domain-derived models with the approximation power of neural dynamics. Despite their growing adoption in biology and engineering, neural augmentation can introduce observational degeneracies that compromise mechanistic identifiability and scientific interpretability. In this paper, we develop a theoretical framework for practical preservation of mechanistic identifiability in HNDEs. We formalize bounded neural correction classes and derive Gronwall-type trajectory and observational discrepancy bounds linking neural perturbations to mechanistic parameter ambiguity. We further establish sufficient conditions under which hybrid neural corrections preserve approximate mechanistic parameter recoverability up to explicitly quantifiable tolerances. Empirical likelihood profile analyses on benchmark systems confirm that neural augmentation systematically weakens--but does not eliminate--mechanistic identifiability, revealing a fundamental expressiveness-identifiability trade-off. These results provide theoretical foundations and actionable guidance for deploying HNDEs in scientific intelligent computing.