A Hybrid Neural Network and Finite Difference Method for Solving Stokes Equations with Singular Forces on an Embedded Interface

Core Concepts

The proposed hybrid method combines the expressive power of neural networks and the convergence of finite difference schemes to efficiently solve Stokes equations with singular forces on an embedded interface in regular domains.

Abstract

The paper presents a hybrid neural-network and finite-difference method for solving Stokes equations with singular forces on an interface in regular domains. The key idea is to decompose the solution into singular and regular parts. The singular part solution is obtained using neural network learning machinery to capture the non-smooth behavior across the interface. The neural networks are trained to satisfy the jump conditions along the interface. The regular part solution is then computed using a traditional MAC (Marker-And-Cell) scheme and Uzawa-type algorithm to solve the resulting Stokes-like system. The regular part solution is smooth across the interface. The overall computational cost is efficient, as it leverages well-established fast Poisson solvers. The method is demonstrated through several numerical experiments in both 2D and 3D, showing second-order convergence for the velocity and first-order convergence for the pressure. The hybrid approach avoids the need for additional discretization efforts near the interface, unlike traditional sharp interface methods.

Stats

The velocity field u = (u1, u2) is given by: u1(r, θ) = (1/8r^2 cos(2θ) + 1/16r^4 cos(4θ) - 1/4r^4 cos(2θ)) for r < 1, and (-1/8r^-2 cos(2θ) + 5/16r^-4 cos(4θ) - 1/4r^-2 cos(4θ)) for r ≥ 1. u2(r, θ) = (-1/8r^2 sin(2θ) + 1/16r^4 sin(4θ) + 1/4r^4 sin(2θ)) for r < 1, and (1/8r^-2 sin(2θ) + 5/16r^-4 sin(4θ) - 1/4r^-2 sin(4θ)) for r ≥ 1. The pressure p(x, y) is given by: x^3 + cos(πx)cos(πy) for r < 1, and cos(πx)cos(πy) for r ≥ 1. The external force g = (g1, g2) is given by: g1(x, y) = (-π sin(πx)cos(πy) + 6x^2 - 3y^2) for r < 1, and (-π sin(πx)cos(πy) - 3(x^4 - 6x^2y^2 + y^4)/(x^2 + y^2)^4) for r ≥ 1. g2(x, y) = (-π sin(πx)cos(πy) - 6xy) for r < 1, and (-π sin(πx)cos(πy) - 12(x^3y - xy^3)/(x^2 + y^2)^4) for r ≥ 1.

Quotes

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Key Insights Distilled From

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

by Che-Chia Cha... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2306.06333.pdf

Deeper Inquiries

How can the proposed hybrid method be extended to handle time-dependent Stokes interface problems, such as those involving vesicle hydrodynamics or electro-hydrodynamics

To extend the proposed hybrid method to handle time-dependent Stokes interface problems, such as those involving vesicle hydrodynamics or electro-hydrodynamics, several modifications and enhancements can be implemented. Time Integration: Incorporate time-stepping schemes like the implicit Euler method or Runge-Kutta methods to evolve the solution in time. This involves updating the solution at each time step based on the previous time step's solution. Dynamic Interface Tracking: Implement algorithms to track the interface's movement over time accurately. This is crucial for problems involving deformable interfaces like vesicles or dynamic interfaces in electro-hydrodynamics. Adaptive Mesh Refinement: Utilize adaptive mesh refinement techniques to dynamically adjust the grid resolution based on the solution's behavior. This can help capture fine details near the interface more efficiently. Incorporating External Forces: Extend the method to handle additional external forces or interactions specific to vesicle hydrodynamics or electro-hydrodynamics, such as electric or magnetic fields. Validation and Verification: Validate the extended method against analytical solutions or experimental data to ensure accuracy and reliability in capturing the time-dependent behavior of the system. By incorporating these enhancements, the hybrid method can effectively tackle time-dependent Stokes interface problems, providing insights into complex phenomena like vesicle dynamics and electro-hydrodynamics.

Can the hybrid method be further improved to achieve higher-order convergence for the pressure field, especially near the corners of the computational domain

Achieving higher-order convergence for the pressure field, especially near the corners of the computational domain, can be challenging but feasible with certain enhancements to the hybrid method: Improved Interpolation: Utilize higher-order interpolation schemes, such as quadratic or cubic interpolation, to approximate the pressure field at the corners more accurately. This can help reduce errors associated with linear interpolation. Local Refinement: Implement local mesh refinement strategies near the corners where the pressure field exhibits higher gradients. This can enhance the resolution in critical regions and improve convergence rates. Boundary Treatment: Develop specialized boundary treatment techniques tailored for the corners to ensure accurate representation of pressure gradients and discontinuities in these areas. Error Analysis: Conduct a detailed error analysis to identify sources of error near the corners and implement corrective measures to mitigate them effectively. By incorporating these strategies, the hybrid method can be refined to achieve higher-order convergence for the pressure field, particularly in challenging regions like the corners of the computational domain.

What are the potential applications and advantages of the hybrid neural network and finite difference approach beyond Stokes interface problems, such as in other types of partial differential equations with discontinuous solutions

The hybrid neural network and finite difference approach proposed for Stokes interface problems can find applications beyond this specific domain in various partial differential equations with discontinuous solutions. Some potential applications and advantages include: Multiphase Flows: The method can be applied to model multiphase flows with complex interfaces, such as fluid-structure interactions, phase separation, or bubble dynamics. The hybrid approach can efficiently capture discontinuities and interface interactions in these systems. Fluid-Structure Interaction: Extending the method to fluid-structure interaction problems can enable the accurate simulation of interactions between fluids and deformable structures, like flexible membranes or moving boundaries. Biomedical Simulations: The approach can be utilized in biomedical simulations involving blood flow, tissue mechanics, or drug delivery, where discontinuous solutions and complex interfaces are prevalent. This can aid in understanding physiological processes and optimizing medical treatments. Material Science: The method can be adapted for simulations in material science, such as phase transitions, solid-fluid interactions, or porous media flow. Its ability to handle discontinuities and interface phenomena makes it suitable for a wide range of material science applications. Advantages: The hybrid method offers a balance between the flexibility of neural networks and the accuracy of finite difference schemes. It provides a robust framework for solving complex PDEs with discontinuous solutions, offering high accuracy, efficiency, and adaptability to various problem domains. By exploring these applications and leveraging the advantages of the hybrid approach, researchers can address a diverse set of challenges in different fields beyond Stokes interface problems.