  Direct numerical simulations (DNS) are accurate but computationally expensive
for predicting materials evolution across timescales, due to the complexity of
the underlying evolution equations, the nature of multiscale spatio-temporal
interactions, and the need to reach long-time integration. We develop a new
method that blends numerical solvers with neural operators to accelerate such
simulations. This methodology is based on the integration of a community
numerical solver with a U-Net neural operator, enhanced by a
temporal-conditioning mechanism that enables accurate extrapolation and
efficient time-to-solution predictions of the dynamics. We demonstrate the
effectiveness of this framework on simulations of microstructure evolution
during physical vapor deposition modeled via the phase-field method. Such
simulations exhibit high spatial gradients due to the co-evolution of different
material phases with simultaneous slow and fast materials dynamics. We
establish accurate extrapolation of the coupled solver with up to 16.5$\times$
speed-up compared to DNS. This methodology is generalizable to a broad range of
evolutionary models, from solid mechanics, to fluid dynamics, geophysics,
climate, and more.
