Towards a Unified Theory of Hybrid Neural Models

Integrating neural networks into mechanistic, equation-based models is a field of growing scientific interest, yet it lacks consistent terminology and a unified mathematical framework. We systematise existing approaches and establish new connections between hybrid modelling, differential equations theory, and deep learning. We clarify the roles of inference, prediction, and generalisation in hybrid models, showing how neural components can capture mathematical structures within and across datasets, and we connect these questions to structural identifiability theory. We further interpret hybrid models through a manifold-learning lens, demonstrating how parametric spaces of missing information can be learned, thereby providing an alternative approach to uncertainty quantification.