A finite operator learning technique for mapping the elastic properties
  of microstructures to their mechanical deformations
Shahed Rezaei Shirko Faroughi Mahdi Asgharzadeh Ali Harandi Gottfried Laschet Stefanie Reese Markus Apel
To develop faster solvers for governing physical equations in solid
mechanics, we introduce a method that parametrically learns the solution to
mechanical equilibrium. The introduced method outperforms traditional ones in
terms of computational cost while acceptably maintaining accuracy. Moreover, it
generalizes and enhances the standard physics-informed neural networks to learn
a parametric solution with rather sharp discontinuities. We focus on
micromechanics as an example, where the knowledge of the micro-mechanical
solution, i.e., deformation and stress fields for a given heterogeneous
microstructure, is crucial. The parameter under investigation is the Young
modulus distribution within the heterogeneous solid system. Our method,
inspired by operator learning and the finite element method, demonstrates the
ability to train without relying on data from other numerical solvers. Instead,
we leverage ideas from the finite element approach to efficiently set up loss
functions algebraically, particularly based on the discretized weak form of the
governing equations. Notably, our investigations reveal that physics-based
training yields higher accuracy compared to purely data-driven approaches for
unseen microstructures. In essence, this method achieves independence from data
and enhances accuracy for predictions beyond the training range. The
aforementioned observations apply here to heterogeneous elastic
microstructures. Comparisons are also made with other well-known operator
learning algorithms, such as DeepOnet, to further emphasize the advantages of
the newly proposed architecture.