Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A hybrid neural-network and MAC scheme for Stokes interface problems

Che-Chia Chang, Chen-Yang Dai, Wei-Fan Hu, Te-Sheng Lin, Ming-Chih Lai

TL;DR

This work tackles Stokes interface problems with singular interfacial forces by a hybrid approach that separates the solution into a singular part learned by neural networks under interface jump constraints and a regular part solved with a traditional MAC finite-difference method. The singular component is trained to satisfy jump conditions, while the regular component is computed via a Stokes-like system discretized on a MAC grid and solved with a Schur-complement, Uzawa-type procedure using fast Poisson solvers. Across 2D and 3D test cases, the method achieves second-order accuracy for velocity and first-order accuracy for pressure, with performance comparable to the immersed interface method and strong robustness to grid refinement. The framework enables easy multi-resolution usage and can be extended to multiple interfaces and time-dependent problems such as vesicle dynamics and electro-hydrodynamics, offering a practical, implementable alternative to sharp-interface corrections.

Abstract

In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity) exhibit non-smooth behaviors across the interface so extra discretization efforts must be paid near the interface in order to have small order of local truncation errors in finite difference schemes. The present hybrid approach avoids such additional difficulty. It combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same time. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular part solution, while the standard MAC scheme is used to obtain the regular part solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy for the velocity and first-order accuracy for the pressure, and it is comparable with the traditional immersed interface method in literature.

A hybrid neural-network and MAC scheme for Stokes interface problems

TL;DR

This work tackles Stokes interface problems with singular interfacial forces by a hybrid approach that separates the solution into a singular part learned by neural networks under interface jump constraints and a regular part solved with a traditional MAC finite-difference method. The singular component is trained to satisfy jump conditions, while the regular component is computed via a Stokes-like system discretized on a MAC grid and solved with a Schur-complement, Uzawa-type procedure using fast Poisson solvers. Across 2D and 3D test cases, the method achieves second-order accuracy for velocity and first-order accuracy for pressure, with performance comparable to the immersed interface method and strong robustness to grid refinement. The framework enables easy multi-resolution usage and can be extended to multiple interfaces and time-dependent problems such as vesicle dynamics and electro-hydrodynamics, offering a practical, implementable alternative to sharp-interface corrections.

Abstract

In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity) exhibit non-smooth behaviors across the interface so extra discretization efforts must be paid near the interface in order to have small order of local truncation errors in finite difference schemes. The present hybrid approach avoids such additional difficulty. It combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same time. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular part solution, while the standard MAC scheme is used to obtain the regular part solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy for the velocity and first-order accuracy for the pressure, and it is comparable with the traditional immersed interface method in literature.
Paper Structure (11 sections, 20 equations, 2 figures, 9 tables)

This paper contains 11 sections, 20 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Hybrid methodology for Stokes interface problems
  3. Singular part solution
  4. Regular part solution
  5. Summary of the hybrid method
  6. Numerical results
  7. Example 1.
  8. Example 2.
  9. Example 3.
  10. Example 4.
  11. Conclusion and future work

Figures (2)

  • Figure 1: The cross-sectional plots of the velocity field and pressure along the grid lines in Example 2. The mesh width is $h=2/N$ with $N=512$.
  • Figure 2: Quiver plot for $(u_1,u_3)$ along the plane $y=h/2$ in Example 4. The mesh width is $h=2/N$ with $N=128$. The color indicates the absolute magnitude of the velocity field.