If it is all about a flexible function to approximate then why don’t we use Fourier Series in place of NN(just for fun of it and experiments, not talking about being mature and solving some practical problem only.)
Its a cool experiment. But its to be noted that NN are good approximations for equations that we cannot solve and that are too complicated. But Fourier Series is solvable with a bunch of known equations. Moreover to prepare the data for this model you will use Fourier series to pass the inputs and get the corresponding outputs as labels; which seems like redundant work.