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<p>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.</p>

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Neural networks for structured grid generation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月11日

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  • Bari Khairullin,
  • Sergey Rykovanov,
  • Rishat Zagidullin

摘要

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.