# Optimal Energy Shaping via Neural Approximators

arXiv: https://arxiv.org/abs/2101.05537 · HTML: https://mlsanity.com/paper/2101.05537/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 14 Jan 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [A Brief Survey of Deep Reinforcement Learning](https://mlsanity.com/paper/1708.05866/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Challenges of Real-World Reinforcement Learning](https://mlsanity.com/paper/1904.12901/) (2019)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Fashionable Modelling with Flux](https://mlsanity.com/paper/1811.01457/) (2018)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Stable Neural Flows](https://mlsanity.com/paper/2003.08063/) (2020)
- [TorchDyn: A Neural Differential Equations Library](https://mlsanity.com/paper/2009.09346/) (2020)
- [Port-Hamiltonian Approach to Neural Network Training](https://mlsanity.com/paper/1909.02702/) (2019)
- [Optimal Control of Port-Hamiltonian Systems: A Time-Continuous Learning Approach](https://mlsanity.com/paper/2007.08645/) (2020)

## Built on by (most cited first, 4 of 4)

- [Total Energy Shaping with Neural Interconnection and Damping Assignment -- Passivity Based Control](https://mlsanity.com/paper/2112.12999/) (2021)
- [Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups](https://mlsanity.com/paper/2401.15107/) (2024)
- [Structured Deep Neural Network-Based Backstepping Trajectory Tracking Control for Lagrangian Systems](https://mlsanity.com/paper/2403.00381/) (2024)
- [Co-Learning Port-Hamiltonian Systems and Optimal Energy-Shaping Control](https://mlsanity.com/paper/2604.26172/) (2026)

## Abstract (arXiv)

We introduce optimal energy shaping as an enhancement of classical passivity-based control methods. A promising feature of passivity theory, alongside stability, has traditionally been claimed to be intuitive performance tuning along the execution of a given task. However, a systematic approach to adjust performance within a passive control framework has yet to be developed, as each method relies on few and problem-specific practical insights. Here, we cast the classic energy-shaping control design process in an optimal control framework; once a task-dependent performance metric is defined, an optimal solution is systematically obtained through an iterative procedure relying on neural networks and gradient-based optimization. The proposed method is validated on state-regulation tasks.