The simulation of multiscale viscoelastic materials poses a significant challenge in computational materials science, requiring expensive numerical solvers that can resolve dynamics of material deformations at the microscopic scale. The theory of homogenization offers an alternative approach to modeling, by locally averaging the strains and stresses of multiscale materials. This procedure eliminates the smaller scale dynamics but introduces a history dependence between strain and stress that proves very challenging to characterize analytically. In the one-dimensional setting, we give the first full characterization of the memory-dependent constitutive laws that arise in multiscale viscoelastic materials. Using this theory, we develop a neural differential equation architecture, that simultaneously across a wide range of material microstructures, accurately predicts their homogenized constitutive laws, thus enabling us to simulate their deformations under forcing. We use the approximation theory of neural operators to provide guarantees on the generalization of our approach to unseen material samples.