Physical networks become what they learn

Physical networks may adapt to have diverse desired functions or properties, whether by design, evolution or learning. The adaptation process is expected to alter the functionality of the network and its own physics. For networks that naturally minimize a physical scalar, e.g. an energy function, adaptation of function is a double optimization problem, minimizing both a physical and a learning cost function. We study how the process of physical adaptation couples the associated two landscapes. In linear systems, such as self-learning resistor networks, we show how adaptation links the physical and learning Hessian matrices, suggesting that the physical responses of the network to perturbations hold much information about the functions it adapted to perform.