# Self-learning Machines based on Hamiltonian Echo Backpropagation

arXiv: https://arxiv.org/abs/2103.04992 · HTML: https://mlsanity.com/paper/2103.04992/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 109 references that are in the map.
- First posted on arXiv 8 Mar 2021.

## Builds on (most cited first, 18 of 18)

- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Photonics for artificial intelligence and neuromorphic computing](https://mlsanity.com/paper/2011.00111/) (2020)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [A Survey of Neuromorphic Computing and Neural Networks in Hardware](https://mlsanity.com/paper/1705.06963/) (2017)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Training of photonic neural networks through in situ backpropagation](https://mlsanity.com/paper/1805.09943/) (2018)
- [Wave Physics as an Analog Recurrent Neural Network](https://mlsanity.com/paper/1904.12831/) (2019)
- [EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations](https://mlsanity.com/paper/2010.07859/) (2020)
- [Theory of neuromorphic computing by waves: machine learning by rogue waves, dispersive shocks, and solitons](https://mlsanity.com/paper/1912.07044/) (2019)
- [Trainable and Dynamic Computing: Error Backpropagation through Physical Media](https://mlsanity.com/paper/1407.6637/) (2014)
- [Superconducting optoelectronic circuits for neuromorphic computing](https://mlsanity.com/paper/1610.00053/) (2016)
- [Photonic Delay Systems as Machine Learning Implementations](https://mlsanity.com/paper/1501.02592/) (2015)
- [Photonic convolutional neural networks using integrated diffractive optics](https://mlsanity.com/paper/2003.12015/) (2020)
- [A superconducting nanowire spiking element for neural networks](https://mlsanity.com/paper/2007.15101/) (2020)
- [Transient chaotic dimensionality expansion by recurrent networks](https://mlsanity.com/paper/2002.11006/) (2020)

## Built on by (most cited first, 28 of 28)

- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Frequency propagation: Multi-mechanism learning in nonlinear physical networks](https://mlsanity.com/paper/2208.08862/) (2022)
- [Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems](https://mlsanity.com/paper/2406.00879/) (2024)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Learning long range dependencies through time reversal symmetry breaking](https://mlsanity.com/paper/2506.05259/) (2025)
- [Ornstein-Uhlenbeck Adaptation as a Mechanism for Learning in Brains and Machines](https://mlsanity.com/paper/2410.13563/) (2024)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Thermodynamic Computing via Autonomous Quantum Thermal Machines](https://mlsanity.com/paper/2308.15905/) (2023)
- [Physical Reservoir Computing Enabled by Solitary Waves and Biologically-Inspired Nonlinear Transformation of Input Data](https://mlsanity.com/paper/2402.03319/) (2024)
- [Local Learning Rules for Out-of-Equilibrium Physical Generative Models](https://mlsanity.com/paper/2506.19136/) (2025)
- [Coercivity and Local Convergence of Physical Learning in Linear Circuits](https://mlsanity.com/paper/2606.15443/) (2026)
- [Noise-based Local Learning using Stochastic Magnetic Tunnel Junctions](https://mlsanity.com/paper/2412.12783/) (2024)
- [OscNet: Machine Learning on CMOS Oscillator Networks](https://mlsanity.com/paper/2502.07192/) (2025)
- [Model-free Optical Processors using In Situ Reinforcement Learning with Proximal Policy Optimization](https://mlsanity.com/paper/2507.05583/) (2025)
- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [SymFlux: deep symbolic regression of Hamiltonian vector fields](https://mlsanity.com/paper/2507.06342/) (2025)
- [Harnessing intuitive local evolution rules for physical learning](https://mlsanity.com/paper/2507.19561/) (2025)
- [Training deep physical neural networks with local physical information bottleneck](https://mlsanity.com/paper/2602.09569/) (2026)
- [Variational quantum algorithm for Gaussian discrete solitons and their boson sampling](https://mlsanity.com/paper/2110.12379/) (2021)
- [Quantum automated learning with provable and explainable trainability](https://mlsanity.com/paper/2502.05264/) (2025)
- [Thermodynamic cost of inference and learning in physical neural networks](https://mlsanity.com/paper/2503.09980/) (2025)
- [Programmable metasurfaces for future photonic artificial intelligence](https://mlsanity.com/paper/2505.11659/) (2025)
- [Laser interferometry as a robust neuromorphic platform for machine learning](https://mlsanity.com/paper/2601.18047/) (2026)
- [Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Model](https://mlsanity.com/paper/2605.21568/) (2026)
- [Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks](https://mlsanity.com/paper/2609.04134/) (2026)
- [Online local learning for generative thermodynamic computing](https://mlsanity.com/paper/2609.15439/) (2026)

## Abstract (arXiv)

A physical self-learning machine can be defined as a nonlinear dynamical system that can be trained on data (similar to artificial neural networks), but where the update of the internal degrees of freedom that serve as learnable parameters happens autonomously. In this way, neither external processing and feedback nor knowledge of (and control of) these internal degrees of freedom is required. We introduce a general scheme for self-learning in any time-reversible Hamiltonian system. We illustrate the training of such a self-learning machine numerically for the case of coupled nonlinear wave fields.