# How to Train Your Resistive Network: Generalized Equilibrium Propagation and Analytical Learning

arXiv: https://arxiv.org/abs/2602.03546 · HTML: https://mlsanity.com/paper/2602.03546/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 11 of 11)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Memristors -- from In-memory computing, Deep Learning Acceleration, Spiking Neural Networks, to the Future of Neuromorphic and Bio-inspired Computing](https://mlsanity.com/paper/2004.14942/) (2020)
- [Memristive Linear Algebra](https://mlsanity.com/paper/2407.20539/) (2024)
- [Uncontrolled learning: co-design of neuromorphic hardware topology for neuromorphic algorithms](https://mlsanity.com/paper/2408.05183/) (2024)
- [Global minimization via classical tunneling assisted by collective force field formation](https://mlsanity.com/paper/2102.03385/) (2021)
- [A unifying approach to self-organizing systems interacting via conservation laws](https://mlsanity.com/paper/2507.02575/) (2025)
- [Self-Organising Memristive Networks as Physical Learning Systems](https://mlsanity.com/paper/2509.00747/) (2025)

## Built on by (most cited first, 2 of 2)

- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)
- [Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning](https://mlsanity.com/paper/2608.30778/) (2026)

## Abstract (arXiv)

Machine learning is a powerful method of extracting meaning from data; unfortunately, current digital hardware is extremely energy-intensive. There is interest in an alternative analog computing implementation that could match the performance of traditional machine learning while being significantly more energy-efficient. However, it remains unclear how to train such analog computing systems while adhering to locality constraints imposed by the physical (as opposed to digital) nature of these systems. Local learning algorithms such as Equilibrium Propagation and Coupled Learning have been proposed to address this issue. In this paper, we develop an algorithm to exactly calculate gradients using a graph theoretic and analytical framework for Kirchhoff's laws. We also introduce Generalized Equilibrium Propagation, a framework encompassing a broad class of Hebbian learning algorithms, including Coupled Learning and Equilibrium Propagation, and show how our algorithm compares. We demonstrate our algorithm using numerical simulations and show that we can train resistor networks without the need for a replica or readout over all resistors, only at the output layer. We also show that under the analytical gradient approach, it is possible to update only a subset of the resistance values without a strong degradation in performance.