  Multiscale problems are widely observed across diverse domains in physics and
engineering. Translating these problems into numerical simulations and solving
them using numerical schemes, e.g. the finite element method, is costly due to
the demand of solving initial boundary-value problems at multiple scales. On
the other hand, multiscale finite element computations are commended for their
ability to integrate micro-structural properties into macroscopic computational
analyses using homogenization techniques. Recently, neural operator-based
surrogate models have shown trustworthy performance for solving a wide range of
partial differential equations. In this work, we propose a hybrid method in
which we utilize deep operator networks for surrogate modeling of the
microscale physics. This allows us to embed the constitutive relations of the
microscale into the model architecture and to predict microscale strains and
stresses based on the prescribed macroscale strain inputs. Furthermore,
numerical homogenization is carried out to obtain the macroscale quantities of
interest. We apply the proposed approach to quasi-static problems of solid
mechanics. The results demonstrate that our constitutive relations-aware
DeepONet can yield accurate solutions even when being confronted with a
restricted dataset during model development.
