High-order expansion of Neural Ordinary Differential Equations flows
This work addresses the opacity of neural ODE dynamics, improving trust for researchers and practitioners in fields like dynamical systems and control, though it is incremental as it builds on existing neural ODE methods.
The paper tackled the problem of limited explainability in neural ordinary differential equations (ODEs) by introducing Event Transition Tensors, a framework based on high-order differentials, which enhanced interpretability and rigour in applications such as prey-predator control models and neural Hamiltonian systems.
Artificial neural networks, widely recognised for their role in machine learning, are now transforming the study of ordinary differential equations (ODEs), bridging data-driven modelling with classical dynamical systems and enabling the development of infinitely deep neural models. However, the practical applicability of these models remains constrained by the opacity of their learned dynamics, which operate as black-box systems with limited explainability, thereby hindering trust in their deployment. Existing approaches for the analysis of these dynamical systems are predominantly restricted to first-order gradient information due to computational constraints, thereby limiting the depth of achievable insight. Here, we introduce Event Transition Tensors, a framework based on high-order differentials that provides a rigorous mathematical description of neural ODE dynamics on event manifolds. We demonstrate its versatility across diverse applications: characterising uncertainties in a data-driven prey-predator control model, analysing neural optimal feedback dynamics, and mapping landing trajectories in a three-body neural Hamiltonian system. In all cases, our method enhances the interpretability and rigour of neural ODEs by expressing their behaviour through explicit mathematical structures. Our findings contribute to a deeper theoretical foundation for event-triggered neural differential equations and provide a mathematical construct for explaining complex system dynamics.