Construction of an artificial neural network for solving the incompressible Navier–Stokes equations

The tasks of analyzing and visualizing the dynamics of viscous incompressible flows of complex geometry based on traditional grid and projection methods are associated with significant requirements for computer performance necessary to achieve the set goals. To reduce the computational load in solving this class of problems, it is possible to apply algorithms for constructing artificial neural networks (ANNs) using exact solutions of the Navier–Stokes equations on a given set of spatial regions as training sets. An ANN is implemented to construct flows in regions that are complexes made up of training sets of standard axisymmetric domains (cylinders, balls, etc.). To reduce the amount of calculations in the case of 3D problems, invariant flow manifolds of lower dimensions are used. This makes it possible to identify the structure of solutions in detail. It is established that typical invariant regions of such flows are figures of rotation, in particular, ones homeomorphic to the torus, which form the structure of a topological bundle, for example, in a ball, cylinder, and general complexes composed of such figures. The structures of flows obtained by approximation based on the simplest 3D unsteady vortex flows are investigated. Classes of exact solutions of the incompressible Navier–Stokes system in bounded regions of $\mathbb{R}_3$ are distinguished based on the superposition of the above-mentioned topological bundles. Comparative numerical experiments suggest that the application of the proposed class of ANNs can significantly speed up the computations, which allows the use of low-performance computers.