  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.
