  To obtain fast solutions for governing physical equations in solid mechanics,
we introduce a method that integrates the core ideas of the finite element
method with physics-informed neural networks and concept of neural operators.
This approach generalizes and enhances each method, learning the parametric
solution for mechanical problems without relying on data from other resources
(e.g. other numerical solvers). We propose directly utilizing the available
discretized weak form in finite element packages to construct the loss
functions algebraically, thereby demonstrating the ability to find solutions
even in the presence of sharp discontinuities. Our focus is on micromechanics
as an example, where knowledge of deformation and stress fields for a given
heterogeneous microstructure is crucial for further design applications. The
primary parameter under investigation is the Young's modulus distribution
within the heterogeneous solid system. Our investigations reveal that
physics-based training yields higher accuracy compared to purely data-driven
approaches for unseen microstructures. Additionally, we offer two methods to
directly improve the process of obtaining high-resolution solutions, avoiding
the need to use basic interpolation techniques. First is based on an
autoencoder approach to enhance the efficiency for calculation on high
resolution grid point. Next, Fourier-based parametrization is utilized to
address complex 2D and 3D problems in micromechanics. The latter idea aims to
represent complex microstructures efficiently using Fourier coefficients.
Comparisons with other well-known operator learning algorithms, further
emphasize the advantages of the newly proposed method.
