We modeled Multilayer Perceptron and Extreme Learning Machine Artificial Neural Networks (ANNs) for computing band structures (BSTs) and photonic band gaps (PBGs) of 2D and 3D photonic crystals (PhCs). We aim at providing fast ANN models which might boost the computations of BDs and PBGs regarding electromagnetic solvers. The case studies considered 2D and 3D PhCs with different lattices, geometries, and materials. Datasets for ANN training were built by varying the geometric shapes' dimensions and the dielectric constants of the case-study PhCs. We demonstrate simple and fast-training ANNs capable of providing accurate BSTs and PGBs through speedy computations.