Finite operator learning for the static analysis of functionally graded microbeams with material heterogeneity

We investigate the static response of two-dimensional functionally graded 2D-FG microbeams using a Finite Operator Learning (FOL) framework. The microbeams exhibit continuously varying material properties in two spatial directions, modeled within the framework of modified couple stress theory. The governing equations are solved using a physics-informed operator learning approach, where a deep feedforward neural network takes spatial coordinates, material gradation, and loading parameters as inputs and approximates the transverse deflection field. The network parameters are trained by minimizing a loss function constructed from the discretized governing equations and boundary conditions. This formulation leads to an unsupervised learning strategy that does not require labeled data from experiments or high-fidelity simulations. By employing finite element shape functions within the discretization process, the method yields an algebraic loss formulation, improving computational efficiency while maintaining consistency with the underlying physics. In addition, the framework enables efficient handling of heterogeneous materials and complex geometries. Once trained, the model can accommodate variations in material properties and loading conditions without retraining, significantly reducing computational cost in parametric studies. The proposed method is validated against available analytical and numerical solutions and the effects of material gradation and loading type on the bending response are systematically analyzed. Furthermore, a comparative study with the Fourier Neural Operator (FNO) is conducted for the 2D-FG microbeam problem. The results indicate that FOL achieves higher accuracy than FNO under identical input–output resolution settings. However, FOL exhibits sensitivity to changes in mesh resolution, leading to reduced performance in cross-resolution evaluations.