# Circuit realization and hardware linearization of monotone operator equilibrium networks

arXiv: https://arxiv.org/abs/2509.13793 · HTML: https://mlsanity.com/paper/2509.13793/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 65 references that are in the map.
- First posted on arXiv 17 Sep 2025.

## Builds on (most cited first, 28 of 28)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Adding Gradient Noise Improves Learning for Very Deep Networks](https://mlsanity.com/paper/1511.06807/) (2015)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Training of photonic neural networks through in situ backpropagation](https://mlsanity.com/paper/1805.09943/) (2018)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [A flexible and fast PyTorch toolkit for simulating training and inference on analog crossbar arrays](https://mlsanity.com/paper/2104.02184/) (2021)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations](https://mlsanity.com/paper/2304.02976/) (2023)
- [A universal approximation theorem for nonlinear resistive networks](https://mlsanity.com/paper/2312.15063/) (2023)
- [A Fast Algorithm to Simulate Nonlinear Resistive Networks](https://mlsanity.com/paper/2402.11674/) (2024)
- [Analog In-memory Training on General Non-ideal Resistive Elements: The Impact of Response Functions](https://mlsanity.com/paper/2502.06309/) (2025)
- [Monotone one-port circuits](https://mlsanity.com/paper/2111.15407/) (2021)
- [Circuit Analysis using Monotone+Skew Splitting](https://mlsanity.com/paper/2211.14010/) (2022)
- [Non-Euclidean Monotone Operator Theory and Applications](https://mlsanity.com/paper/2303.11273/) (2023)
- [Learning in memristive electrical circuits](https://mlsanity.com/paper/2409.15136/) (2024)
- [Optimization Algorithm Design via Electric Circuits](https://mlsanity.com/paper/2411.02573/) (2024)
- [Convergence of energy-based learning in linear resistive networks](https://mlsanity.com/paper/2503.00349/) (2025)
- [KirchhoffNet: A Scalable Ultra Fast Analog Neural Network](https://mlsanity.com/paper/2310.15872/) (2023)
- [Arbitrarily Small Execution-Time Certificate: What was Missed in Analog Optimization](https://mlsanity.com/paper/2505.10366/) (2025)
- [Operator-Splitting Methods for Neuromorphic Circuit Simulation](https://mlsanity.com/paper/2505.22363/) (2025)

## Built on by (most cited first, 1 of 1)

- [Quantization Robustness of Monotone Operator Equilibrium Networks](https://mlsanity.com/paper/2603.10562/) (2026)

## Abstract (arXiv)

It is shown that the port behavior of a resistor-diode network corresponds to the solution of a ReLU monotone operator equilibrium network (a neural network in the limit of infinite depth), giving a parsimonious construction of a neural network in analog hardware. We furthermore show that the gradient of such a circuit can be computed directly in hardware, using a procedure we call hardware linearization. This allows the network to be trained in hardware, which we demonstrate with a device-level circuit simulation. We extend the results to cascades of resistor-diode networks, which can be used to implement feedforward and other asymmetric networks. We finally show that different nonlinear elements give rise to different activation functions, and introduce the novel diode ReLU which is induced by a non-ideal diode model.