This research introduces physics of neural networks which are able to be trained for the purpose of solving supervised learning duties with considering any provided physics laws explained by the dynamic problem. In this research, we report developments in the concept of solving two key types of problems: data-driven discovery and solution of vibrations. Using the method of variation, the governing equations pertaining two-directional functionally graded nanobeams have been obtained by utilizing the discrete singular convolution integrated method (DSC-IM). A nonlocal strain/stress gradient theory (NS/SGT) is initiated in this study, which is a sized dependent theory. Made of concrete material the material is varied in both transverse and axial directions. The role of FG power index in both directions, length/width ratio, and size-dependent factor, then, has been investigated on the frequency of studied nanobeams by performing a parametric study. Comparing the results with already reported findings elsewhere, validation of the present approach is verified. The findings of the present study would be helpful for the design of nanostructures made of concretes in Standard texts or handbooks.