# Perturbative Contrastive Physical Learning

arXiv: https://arxiv.org/abs/2606.09756 · HTML: https://mlsanity.com/paper/2606.09756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Learning by non-interfering feedback chemical signaling in physical networks](https://mlsanity.com/paper/2203.12098/) (2022)
- [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)
- [The hardware is the software](https://mlsanity.com/paper/2310.18335/) (2023)
- [Training all-mechanical neural networks for task learning through in situ backpropagation](https://mlsanity.com/paper/2404.15471/) (2024)

## Abstract (arXiv)

Responses to perturbations are key to understanding physical systems. The ability to contrast such responses by comparing how a system reacts under slightly different conditions provides a mechanism for learning. Here, we introduce Perturbative Contrastive Physical Learning (PCPL), a general framework in which learning emerges from measurable contrasts between physical states produced by controlled changes to inputs, boundary conditions, parameters, or interpreter functions. PCPL unifies and extends prior approaches: Equilibrium Propagation is rooted in contrasts between free and nudged equilibria in energy-based systems, while Frequency Propagation corresponds to contrasts extracted from sinusoidally driven, frequency-demodulated responses. We show that contrast-driven updates can reflect either local sensitivities or global inverse-problem structure, yet do not require centralized gradient computation. Instead, effective learning geometry emerges implicitly from the system's own physical response, allowing learning behavior to arise without an external processor or explicit backpropagation. We demonstrate PCPL in two platforms: (i) spring networks that update bond stiffness using measured displacements and forces, and (ii) continuous-variable photonic circuits trained via x quadrature measurements and finite-difference estimates of the Jacobian. Both platforms successfully learn classification tasks. We further show that a continuous-variable photonic circuit can be trained to implement analog multiplication, illustrating a step toward more autonomous physical learning systems.