It has been told that deep learning is a black box. The universal approximation theorem was the key theorem which makes the stories going on. On the other hand, in the perspective of the function classes generated by deep neural network, it can be analyzed by in terms of the choice of the various activation functions. The piecewise linear functions, fourier series, wavelets and many other classes would be considered for the purpose of the tasks such as classification, prediction and generation models which heavily depend on the data sets. It might be a challenging problem for mathematicians to develop a new optimization theory depending on the various function classes.

For a given stable subalgebra of the formal power series ring, its Laurent extension, or others, we define an operator algebra over the subalgebra. One of the important operator algebras is the Weyl algebra or its generalization. We define generalized radical Weyl algebras (GRWA) and define the generalized radical Weyl algebra modules, we prove that the algebras and modules are simple. An automorphism of the GRWA define a twisted simple module as well. Since GRWA is an associative algebra, it has an ${\Bbb F}$-subalgebra which is a Lie algebra with respect to the commutator and we show that the Lie algebra is simple. We consider some other generalized Weyl algebra and its descended consequences as well.