On the construction of an artificial neural network for solving a system of equations Navier–Stokes in the case of incompressible fluid

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Abstract
The tasks of analyzing and visualizing the dynamics of a viscous incompressible fluid in conditions of complex flow geometry based on traditional grid and projection methods are associated with significant requirements for computer performance to achieve the set goals. To reduce the computational load in solving this class of problems, algorithms for constructing artificial neural networks (ANNs) can be used, using exact solutions of the Navier–Stokes equation system on a given set of spatial regions as training sets. An ANN is implemented to construct flows in areas that are complexes made up of training sets of standard axisymmetric regions (cylinders, balls, etc.). To reduce the amount of calculations in the case of 3-D problems, invariant flow manifolds with smaller dimensions are used. This allows you to identify the detailed structure of solutions. It is established that the typical invariant regions of such flows are rotation figures, in particular, homeomorphic torus, forming the structure of a topological bundle, for example, in a ball, a cylinder and in general complexes composed of such figures. The structures of the flows obtained by approximation by the simplest 3-D vortex unsteady flows are investigated. Classes of exact solutions of the Navier–Stokes system for an incompressible fluid in bounded regions of space based on the superposition of the above topological bundles are distinguished. Comparative computational experiments indicate a significant acceleration of computational work in the case of using the proposed class of ANNs, which allows the use of computing equipment with low performance.