A universal approximation theorem for nonlinear resistive networks

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.