Physics-informed hybrid learning for bending analysis of multidirectional porous FG shells

The bending response of metal foam shells is dictated by a complex interplay of geometric curvature and graded porosity, which complicates accurate modeling. To address this issue, we create a thorough Physics-Informed (PI) Machine Learning framework that integrates an improved higher-order shear deformation theory with a fi finite element formulation to forecast the bending behavior of multidirectional functionally graded (MFG) porous metal foam doubly curved shells. A four-node quadrilateral element is developed to model both hard-core and soft-core architectures with fi five different porosity distribution patterns. The model's predictions match up very well with the benchmark results, which shows that it is numerically strong across a wide range of geometries and boundary conditions. Five hybrid machine learning surrogate models, PI-ANN + GBM, PI-ANN + RF, PI-ANN + DNN, PI-ANN + DT, and PI-ANN + XGBoost are trained on simulation data to predict displacements and stresses. Among them, PI-ANN + XGBoost and PI-ANN + DNN achieve the highest predictive accuracy, with R2 values exceeding 0.98 and the lowest RMSE values. These models effectively capture the nonlinear stress–strain relationships while maintaining physical consistency through embedded physics constraints. Overall, the proposed framework modeling exhibits advanced hybrid learning to make better predictions and make predictions that are physically accurate.