Neural networks for structured grid generation

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Geometric Modeling

Background:

• Numerical solutions of partial differential equations (PDEs) are simplified on regular domains.
• Complex geometries often require specialized coordinate systems for efficient computation.
• Existing methods for body-fitted coordinate (BFC) generation can be cumbersome and lack differentiability.

Purpose of the Study:

• To develop a novel neural network (NN)-based approach for generating 2D body-fitted curvilinear coordinate systems (BFCs).
• To enable numerical solutions on regular grids even for complex geometries.
• To provide a differentiable mapping for BFCs, allowing exact Jacobian computation.

Main Methods:

• A feed-forward neural network (FNN) is employed as a geometric transformation to represent a diffeomorphism.
• The FNN is trained using an optimization system analogous to physics-informed neural network (PINN) solutions of Winslow equations.
• The approach focuses on creating BFCs that map complex physical domains to regular computational grids.

Main Results:

• The proposed FNN-based method successfully generates 2D BFCs for complex geometries.
• The generated mapping is differentiable, allowing for exact calculation of Jacobian matrices.
• Interior node distribution can be modified without regenerating the entire mapping, offering flexibility.

Conclusions:

• The NN-based approach offers a flexible and efficient alternative for generating BFCs.
• This method simplifies the numerical solution of PDEs on complex domains by leveraging regular grids.
• The exact differentiability and adaptability of the FNN mapping represent significant advantages over classical BFC generation techniques.