🤖 AI Summary
Analog circuits composed of voltage sources, linear resistors, ideal diodes, and voltage-controlled voltage sources (VCVSs) lack a rigorous theoretical foundation for universal function approximation.
Method: We propose a structured mapping that establishes an exact equivalence between ReLU neural networks and such nonlinear analog circuits under ideal-component assumptions.
Contribution/Results: We provide the first rigorous proof that these circuits can approximate any continuous function to arbitrary precision, thereby establishing their computational completeness. This result furnishes a solid theoretical basis for self-learning analog neural networks. Furthermore, the proposed circuit architecture is compatible with Equilibrium Propagation—a biologically plausible training framework—and supports end-to-end differentiable analog hardware implementation. By bridging the gap between analog circuit theory and neural computation, our work overcomes the longstanding limitation that analog circuit representational capacity lacks formal theoretical guarantees.
📝 Abstract
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.
