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Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

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Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups.

Yannik P Wotte¹, Federico Califano¹, Stefano Stramigioli¹

¹Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands.

Entropy (Basel, Switzerland)

|August 28, 2025

Summary

Neural ordinary differential equations (neural ODEs) are extended to differentiable manifolds and Lie groups, enabling optimization of complex dynamical systems beyond standard Euclidean spaces.

Keywords:

Lie groups differential geometry machine learning neural ordinary differential equations optimal control

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Area of Science:

Dynamical Systems
Machine Learning
Differential Geometry

Background:

Neural ordinary differential equations (neural ODEs) are widely used for parameter optimization in dynamical systems.
Existing theoretical results for neural ODEs are primarily focused on Euclidean space (Rn).
Many real-world dynamical systems evolve on more complex spaces like differentiable manifolds and Lie groups.

Purpose of the Study:

To extend the theoretical framework of neural ODEs to differentiable manifolds.
To provide a unifying derivation of existing results for manifold neural ODEs.
To introduce extensions of neural ODEs for systems evolving on Lie groups.

Main Methods:

Collection and unification of recent theoretical results for neural ODEs on manifolds.
Development of a tutorial framework for extending existing neural ODE methods to differentiable manifolds.
Extension of manifold neural ODEs to exploit the structure of Lie groups.

Main Results:

A unified derivation of neural ODEs on differentiable manifolds is presented.
The framework is shown to be applicable to various dynamical systems.
A novel extension of manifold neural ODEs is developed for Lie groups, leveraging their specific structure.

Conclusions:

Neural ODEs can be effectively generalized to differentiable manifolds and Lie groups.
This generalization expands the applicability of neural ODEs to a wider range of complex dynamical systems.
The presented methods offer a pathway for future research in geometric deep learning and physics-informed neural networks.