# A universal approximation theorem for nonlinear resistive networks

arXiv: https://arxiv.org/abs/2312.15063 · HTML: https://mlsanity.com/paper/2312.15063/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 22 Dec 2023.

## Builds on (most cited first, 12 of 12)

- [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)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Backpropagation-free Training of Deep Physical Neural Networks](https://mlsanity.com/paper/2304.11042/) (2023)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Nonlinear optical encoding enabled by recurrent linear scattering](https://mlsanity.com/paper/2307.08558/) (2023)
- [Frequency propagation: Multi-mechanism learning in nonlinear physical networks](https://mlsanity.com/paper/2208.08862/) (2022)
- [Trainable and Dynamic Computing: Error Backpropagation through Physical Media](https://mlsanity.com/paper/1407.6637/) (2014)
- [Desynchronous Learning in a Physics-Driven Learning Network](https://mlsanity.com/paper/2201.04626/) (2022)

## Built on by (most cited first, 6 of 6)

- [A Fast Algorithm to Simulate Nonlinear Resistive Networks](https://mlsanity.com/paper/2402.11674/) (2024)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (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)
- [Laser interferometry as a robust neuromorphic platform for machine learning](https://mlsanity.com/paper/2601.18047/) (2026)
- [An all-transistor physical-learning transformer can be trained to perform in-context learning](https://mlsanity.com/paper/2610.12571/) (2026)

## Abstract (arXiv)

Resistor networks have recently been studied as analog computing platforms for machine learning, particularly due to their compatibility with the Equilibrium Propagation training framework. In this work, we explore the computational capabilities of these networks. We prove that electrical networks consisting of voltage sources, linear resistors, diodes, and voltage-controlled voltage sources (VCVSs) can approximate any continuous function to arbitrary precision. Central to our proof is a method for translating a neural network with rectified linear units into an approximately equivalent electrical network comprising these four elements. Our proof relies on two assumptions: (a) that circuit elements are ideal, and (b) that variable resistor conductances and VCVS amplification factors can take any value (arbitrarily small or large). Our findings provide insights that could guide the development of universal self-learning electrical networks.