  Multiscale modeling is an effective approach for investigating multiphysics
systems with largely disparate size features, where models with different
resolutions or heterogeneous descriptions are coupled together for predicting
the system's response. The solver with lower fidelity (coarse) is responsible
for simulating domains with homogeneous features, whereas the expensive
high-fidelity (fine) model describes microscopic features with refined
discretization, often making the overall cost prohibitively high, especially
for time-dependent problems. In this work, we explore the idea of multiscale
modeling with machine learning and employ DeepONet, a neural operator, as an
efficient surrogate of the expensive solver. DeepONet is trained offline using
data acquired from the fine solver for learning the underlying and possibly
unknown fine-scale dynamics. It is then coupled with standard PDE solvers for
predicting the multiscale systems with new boundary/initial conditions in the
coupling stage. The proposed framework significantly reduces the computational
cost of multiscale simulations since the DeepONet inference cost is negligible,
facilitating readily the incorporation of a plurality of interface conditions
and coupling schemes. We present various benchmarks to assess accuracy and
speedup, and in particular we develop a coupling algorithm for a time-dependent
problem, and we also demonstrate coupling of a continuum model (finite element
methods, FEM) with a neural operator representation of a particle system
(Smoothed Particle Hydrodynamics, SPH) for a uniaxial tension problem with
hyperelastic material. What makes this approach unique is that a well-trained
over-parametrized DeepONet can generalize well and make predictions at a
negligible cost.
