Bridging mechanistic modeling and machine learning: identifiability and uncertainty in hybrid models for computational biology

Mechanistic models based on ordinary differential equations represent one of the main mathematical frameworks for describing the dynamics of biological systems. They are formulated by mathematically encoding the available biological knowledge of a system into equations governing its dynamics. Despite their interpretability and predictive capabilities, developing such models is particularly challenging because it requires a sufficiently detailed understanding of the underlying biological mechanisms. In real-world applications, however, biological knowledge is often incomplete or uncertain, making the construction of fully mechanistic models difficult or even impossible. Recent advances in machine learning have provided a promising solution to this limitation through Hybrid Neural Ordinary Differential Equations (HNODEs), in which the known components of the system are described mechanistically while the unknown dynamics are represented by neural networks. Despite the potential of this framework, several methodological tools that are well established for fully mechanistic models remain largely unexplored in the context of HNODEs. This thesis addresses two of these methodological challenges through two complementary contributions. The first contribution concerns the calibration of HNODE models and the assessment of the identifiability of the parameters in their mechanistic component. In HNODEs, model calibration requires the simultaneous estimation of both the parameters of the mechanistic component and those of the neural networks approximating the unknown dynamics. While the latter are typically overparameterized and do not possess a direct biological interpretation, the parameters in the mechanistic component are biologically meaningful, and their values should be both accurately estimated and identifiable from the available experimental data. Building upon state-of-the-art methodologies from machine learning and mechanistic modeling, we propose an end-to-end pipeline for the simultaneous estimation of all model parameters together with the assessment of the practical identifiability of the parameters in the mechanistic component. The second contribution addresses the problem of uncertainty quantification in HNODE predictions. The flexibility introduced by neural networks enables HNODEs to represent complex and partially unknown biological processes, but also raises fundamental questions regarding the reliability of their predictions, particularly when models are used to extrapolate beyond the experimental conditions available during calibration. To address this challenge, we develop a novel uncertainty quantification methodology that combines concepts from mechanistic modeling and machine learning to provide more reliable estimates of predictive uncertainty in scenarios different from those used to calibrate the model. The proposed methodologies are validated on a collection of in silico benchmark problems representative of challenges commonly encountered in computational biology. Although the two contributions address different methodological questions, they are deeply connected, as both rely on the local geometric analysis of the calibration objective function. Overall, this thesis contributes new methodological tools for working with HNODE models, supporting their adoption in biology.