# Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial

arXiv: https://arxiv.org/abs/2311.00537 · HTML: https://mlsanity.com/paper/2311.00537/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 50 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors](https://mlsanity.com/paper/2207.02696/) (2022)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Survey of Machine Learning Accelerators](https://mlsanity.com/paper/2009.00993/) (2020)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Gpt-4: A Review on Advancements and Opportunities in Natural Language Processing](https://mlsanity.com/paper/2305.03195/) (2023)
- [Desynchronous Learning in a Physics-Driven Learning Network](https://mlsanity.com/paper/2201.04626/) (2022)

## Built on by (most cited first, 20 of 20)

- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [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)
- [A universal approximation theorem for nonlinear resistive networks](https://mlsanity.com/paper/2312.15063/) (2023)
- [Nonlinear thermodynamic computing out of equilibrium](https://mlsanity.com/paper/2412.17183/) (2024)
- [Local Learning Rules for Out-of-Equilibrium Physical Generative Models](https://mlsanity.com/paper/2506.19136/) (2025)
- [Near-Equilibrium Propagation training in nonlinear wave systems](https://mlsanity.com/paper/2510.16084/) (2025)
- [Equilibrium Propagation for Non-Conservative Systems](https://mlsanity.com/paper/2602.03670/) (2026)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (2026)
- [How to Train Your Resistive Network: Generalized Equilibrium Propagation and Analytical Learning](https://mlsanity.com/paper/2602.03546/) (2026)
- [A Conservation Law for Equilibrium Propagation and Coupled Learning](https://mlsanity.com/paper/2606.15444/) (2026)
- [Circuit realization and hardware linearization of monotone operator equilibrium networks](https://mlsanity.com/paper/2509.13793/) (2025)
- [Scaling of hardware-compatible perturbative training algorithms](https://mlsanity.com/paper/2501.15403/) (2025)
- [Physical Reinforcement Learning](https://mlsanity.com/paper/2511.17789/) (2025)
- [Analog Physical Systems Can Exhibit Double Descent](https://mlsanity.com/paper/2511.17825/) (2025)
- [The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks](https://mlsanity.com/paper/2604.10272/) (2026)
- [Perturbative Contrastive Physical Learning](https://mlsanity.com/paper/2606.09756/) (2026)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)
- [Physical Muon: Orthogonalization as an Equilibrium Computation](https://mlsanity.com/paper/2609.37525/) (2026)
- [An all-transistor physical-learning transformer can be trained to perform in-context learning](https://mlsanity.com/paper/2610.12571/) (2026)

## Abstract (arXiv)

Standard deep learning algorithms require differentiating large nonlinear networks, a process that is slow and power-hungry. Electronic learning metamaterials offer potentially fast, efficient, and fault-tolerant hardware for analog machine learning, but existing implementations are linear, severely limiting their capabilities. These systems differ significantly from artificial neural networks as well as the brain, so the feasibility and utility of incorporating nonlinear elements have not been explored. Here we introduce a nonlinear learning metamaterial -- an analog electronic network made of self-adjusting nonlinear resistive elements based on transistors. We demonstrate that the system learns tasks unachievable in linear systems, including XOR and nonlinear regression, without a computer. We find our nonlinear learning metamaterial reduces modes of training error in order (mean, slope, curvature), similar to spectral bias in artificial neural networks. The circuitry is robust to damage, retrainable in seconds, and performs learned tasks in microseconds while dissipating only picojoules of energy across each transistor. This suggests enormous potential for fast, low-power computing in edge systems like sensors, robotic controllers, and medical devices, as well as manufacturability at scale for performing and studying emergent learning.