Training capacity of viscoelastic materials and failure through a dynamical transition

Disordered materials can be endowed with specific elastic responses by applying a select set of training strains. The ensuing internal stresses cause plastic deformations that alter the microscopic structure, and evolve the system towards the desired elastic response. We study the complexity of responses that can be attained, as expressed in the number of sites that are simultaneously controlled. With increased complexity convergence becomes very slow. The training error decays as power-law, with an exponent that varies continuously and appears to vanish at a critical threshold. We argue that this is a dynamical transition, reminiscent of the Griffiths transition in absorbing state problems. We also study the capacity as a function of system size. Our results suggest that the capacity is extensive for near isostatic networks, and sub-extensive for over-coordinated networks. Our work explains how the presence of an exotic critical point affects the convergence of training, and may be relevant to understanding learning in physical systems.