mlsanityData to 7 Oct 2026

Self-learning Machines based on Hamiltonian Echo Backpropagation

arXiv 2103.04992 ↗First posted 8 Mar 2021cs.LG nlin.AO physics.data-ancited by 28

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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.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Builds on 18 of its 109 references that are in the map. First posted on arXiv 8 Mar 2021.

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18 of 109 references.

References · 18

  1. 2018Invertible Residual Networkscited by 364
  2. 2016Exponential expressivity in deep neural networks through transient chaoscited by 260
  3. 2016Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagationcited by 251
  4. 2017The Reversible Residual Network: Backpropagation Without Storing Activationscited by 215
  5. 2020Photonics for artificial intelligence and neuromorphic computingcited by 111
  6. 2018Dendritic cortical microcircuits approximate the backpropagation algorithmcited by 96
  7. 2017A Survey of Neuromorphic Computing and Neural Networks in Hardwarecited by 82
  8. 2017Reversible Architectures for Arbitrarily Deep Residual Neural Networkscited by 69
  9. 2018Training of photonic neural networks through in situ backpropagationcited by 45
  10. 2019Wave Physics as an Analog Recurrent Neural Networkcited by 42
  11. 2020EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementationscited by 29
  12. 2019Theory of neuromorphic computing by waves: machine learning by rogue waves, dispersive shocks, and solitonscited by 19
6 more
  1. 2014Trainable and Dynamic Computing: Error Backpropagation through Physical Mediacited by 9
  2. 2016Superconducting optoelectronic circuits for neuromorphic computingcited by 6
  3. 2015Photonic Delay Systems as Machine Learning Implementationscited by 5
  4. 2020Photonic convolutional neural networks using integrated diffractive opticscited by 4
  5. 2020A superconducting nanowire spiking element for neural networkscited by 3
  6. 2020Transient chaotic dimensionality expansion by recurrent networkscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 1; 2023: 2; 2024: 6; 2025: 10; 2026: 7.

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  1. 2021Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systemscited by 125
  2. 2024Training of Physical Neural Networkscited by 47
  3. 2023Training an Ising Machine with Equilibrium Propagationcited by 29
  4. 2022Frequency propagation: Multi-mechanism learning in nonlinear physical networkscited by 11
  5. 2024Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systemscited by 9
  6. 2024Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocitycited by 9
  7. 2024Ornstein-Uhlenbeck Adaptation as a Mechanism for Learning in Brains and Machinescited by 6
  8. 2025Equilibrium Propagation for Learning in Lagrangian Dynamical Systemscited by 6
  9. 2025Learning long range dependencies through time reversal symmetry breakingcited by 6
  10. 2023Thermodynamic Computing via Autonomous Quantum Thermal Machinescited by 5
  11. 2024Physical Reservoir Computing Enabled by Solitary Waves and Biologically-Inspired Nonlinear Transformation of Input Datacited by 5
  12. 2024Noise-based Local Learning using Stochastic Magnetic Tunnel Junctionscited by 2
16 more
  1. 2025OscNet: Machine Learning on CMOS Oscillator Networkscited by 2
  2. 2025Local Learning Rules for Out-of-Equilibrium Physical Generative Modelscited by 2
  3. 2025Model-free Optical Processors using In Situ Reinforcement Learning with Proximal Policy Optimizationcited by 2
  4. 2026Coercivity and Local Convergence of Physical Learning in Linear Circuitscited by 2
  5. 2025SymFlux: deep symbolic regression of Hamiltonian vector fieldscited by 1
  6. 2025Harnessing intuitive local evolution rules for physical learningcited by 1
  7. 2026A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computingcited by 1
  8. 2021Variational quantum algorithm for Gaussian discrete solitons and their boson samplingcited by 0
  9. 2025Quantum automated learning with provable and explainable trainabilitycited by 0
  10. 2025Thermodynamic cost of inference and learning in physical neural networkscited by 0
  11. 2025Programmable metasurfaces for future photonic artificial intelligencecited by 0
  12. 2026Laser interferometry as a robust neuromorphic platform for machine learningcited by 0
  13. 2026Training deep physical neural networks with local physical information bottleneckcited by 0
  14. 2026Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Modelcited by 0
  15. 2026Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networkscited by 0
  16. 2026Online local learning for generative thermodynamic computingcited by 0