The macroscopic properties of materials that we observe and exploit in engineering application result from complex interactions between physics at multiple length and time scales: electronic, atomistic, defects, domains etc. Multiscale modeling seeks to understand these interactions by exploiting the inherent hierarchy where the behavior at a coarser scale regulates and averages the behavior at a finer scale. This requires the repeated solution of computationally expensive finer-scale models, and often a priori knowledge of those aspects of the finer-scale behavior that affect the coarser scale (order parameters, state variables, descriptors, etc.). We examine the use of machine-learning, and especially deep neural networks, to harness data generated by repeatedly solving the finer scale model to: (i) gain insights into the history dependence and the macroscopic internal variables that govern the overall response; and (ii) to create a computationally efficient surrogate of its solution operator, that can directly be used at the coarser scale with no further modeling. We do so by introducing a recurrent neural operator (RNO), and show that: (i) the architecture and the learned internal variables can provide insight into the physics of the macroscopic problem; and (ii) that the RNO can provide multiscale, specifically FE^2, accuracy at a cost comparable to a conventional empirical constitutive relation.