  Interacting many-body physical systems ranging from neural networks in the
brain to folding proteins to self-modifying electrical circuits can learn to
perform diverse tasks. This learning, both in nature and in engineered systems,
can occur through evolutionary selection or through dynamical rules that drive
active learning from experience. Here, we show that \added{learning in linear
physical networks with weak input signals} leaves architectural imprints on the
Hessian of a physical system. Compared to a generic organization of the system
components, (a) the effective physical dimension of the response to inputs
decreases, (b) the response of physical degrees of freedom to random
perturbations (or system ``susceptibility'') increases, and (c) the
low-eigenvalue eigenvectors of the Hessian align with the task. Overall, these
effects embody the typical scenario for learning processes in physical systems
in the weak input regime, suggesting ways of discovering whether a physical
network may have been trained.
