Neural networks for structured grid generation.

Abstract

Numerical solutions of partial differential equations (PDEs) on regular domains provide simplicity as we can rely on the structure of the space. We investigate a novel neural network (NN) - based approach to generate 2-dimensional body-fitted curvilinear coordinate systems (BFCs) that allow to stay on regular grids even when the complex geometry is considered. We describe a feed-forward neural network (FNN) as a geometric transformation that can represent a diffeomorphism under certain constraints and approximations, followed by the ways of training it to create BFCs. We show that the optimization system is similar to a physics informed neural network (PINN) based solution of Winslow equations. Unlike in classical BFC generation, FNN provides a differentiable mapping between spaces, and all the Jacobian matrices may be obtained exactly at any given point. Also, it allows to change an interior nodes distribution without the need of recreating the whole mapping.