Axial Symmetric Navier Stokes Equations and the Beltrami /anti Beltrami spectrum in view of Physics Informed Neural Networks

In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply reprised and developed in 2022 with M. Trigiante. Instead of a compact $3$-torus $T^3=\mathbb{R}^3/Λ$ where $Λ$ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic $1$-forms that, for each energy level, decomposes into six components: two Beltrami, two anti-Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an $L^2$ space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication.