Physical networks become what they learn

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.