Physical networks become what they learn

TL;DR

This paper investigates how electrical networks adapt their conductances to learn desired responses by coupling cost function minimization with power dissipation, revealing how physical responses encode learned functions.

Contribution

It introduces a framework linking the cost landscape and physical power dissipation in adaptive electrical networks, highlighting the role of the physical response in learning.

Findings

Adaptation couples the cost and physical Hessian matrices.

Physical responses encode information about learned functions.

The study reveals how physical and cost landscapes interact in network learning.

Abstract

Physical networks can develop diverse responses, or functions, by design, evolution or learning. We focus on electrical networks of nodes connected by resistive edges. Such networks can learn by adapting edge conductances to lower a cost function that penalizes deviations from a desired response. The network must also satisfy Kirchhoff's law, balancing currents at nodes, or, equivalently, minimizing total power dissipation by adjusting node voltages. The adaptation is thus a double optimization process, in which a cost function is minimized with respect to conductances, while dissipated power is minimized with respect to node voltages. Here we study how this physical adaptation couples the cost landscape, the landscape of the cost function in the high-dimensional space of edge conductances, to the physical landscape, the dissipated power in the high-dimensional space of node voltages. We show how adaptation links the physical and cost Hessian matrices, suggesting that the physical response of networks to perturbations holds significant information about the functions to which they are adapted.