# Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups

arXiv: https://arxiv.org/abs/2401.15107 · HTML: https://mlsanity.com/paper/2401.15107/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 71 references that are in the map.
- First posted on arXiv 25 Jan 2024.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [How to Train Your Robot with Deep Reinforcement Learning; Lessons We've Learned](https://mlsanity.com/paper/2102.02915/) (2021)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Deep Learning on Lie Groups for Skeleton-based Action Recognition](https://mlsanity.com/paper/1612.05877/) (2016)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [Active Learning in Robotics: A Review of Control Principles](https://mlsanity.com/paper/2106.13697/) (2021)
- [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023)
- [Machine Learning Lie Structures & Applications to Physics](https://mlsanity.com/paper/2011.00871/) (2020)
- [Port-Hamiltonian Neural ODE Networks on Lie Groups For Robot Dynamics Learning and Control](https://mlsanity.com/paper/2401.09520/) (2024)
- [Adaptive Control of SE(3) Hamiltonian Dynamics with Learned Disturbance Features](https://mlsanity.com/paper/2109.09974/) (2021)
- [Lie Algebraic Cost Function Design for Control on Lie Groups](https://mlsanity.com/paper/2204.09177/) (2022)
- [Optimal Energy Shaping via Neural Approximators](https://mlsanity.com/paper/2101.05537/) (2021)
- [Geometric Nonlinear PID Control of a Quadrotor UAV on SE(3)](https://mlsanity.com/paper/1304.6765/) (2013)
- [Discovering Efficient Periodic Behaviours in Mechanical Systems via Neural Approximators](https://mlsanity.com/paper/2212.14253/) (2022)

## Abstract (arXiv)

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.