# High-order expansion of Neural Ordinary Differential Equations flows

arXiv: https://arxiv.org/abs/2504.08769 · HTML: https://mlsanity.com/paper/2504.08769/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 52 references that are in the map.
- First posted on arXiv 2 Apr 2025.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [TorchDyn: A Neural Differential Equations Library](https://mlsanity.com/paper/2009.09346/) (2020)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [On the stability analysis of deep neural network representations of an optimal state-feedback](https://mlsanity.com/paper/1812.02532/) (2018)
- [Aggressive Online Control of a Quadrotor via Deep Network Representations of Optimality Principles](https://mlsanity.com/paper/1912.07067/) (2019)
- [Neural representation of a time optimal, constant acceleration rendezvous](https://mlsanity.com/paper/2203.15490/) (2022)
- [Revisiting high-order Taylor methods for astrodynamics and celestial mechanics](https://mlsanity.com/paper/2105.00800/) (2021)
- [Guidance and Control Networks with Periodic Activation Functions](https://mlsanity.com/paper/2405.18084/) (2024)
- [Guidance & Control Networks for Time-Optimal Quadcopter Flight](https://mlsanity.com/paper/2305.02705/) (2023)
- [Reliable event detection for Taylor methods in astrodynamics](https://mlsanity.com/paper/2204.09948/) (2022)
- [Optimality Principles in Spacecraft Neural Guidance and Control](https://mlsanity.com/paper/2305.13078/) (2023)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Conditional Latent ODEs for Motion Prediction in Autonomous Driving](https://mlsanity.com/paper/2405.19183/) (2024)

## Built on by (most cited first, 1 of 1)

- [Comparing Behavioural Cloning and Reinforcement Learning for Spacecraft Guidance and Control Networks](https://mlsanity.com/paper/2507.19535/) (2025)

## Abstract (arXiv)

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.