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Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations

Yunlong Li, Fei Wang, Lingxiao Li

TL;DR

A Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions, and provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

Abstract

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional numerical solvers. To overcome these difficulties, we propose a Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions. Unlike deep neural network (DNN) approaches that rely on expensive nonlinear and nonconvex optimization, SP-RaNN reformulates the training process into a linear least-squares system, thereby eliminating nonconvex optimization. The method linearizes the governing equations through Picard or Newton iterations, discretizes them at collocation points within the domain and on the boundaries using finite-difference schemes, and solves the resulting linear system via a linear least-squares procedure. By design, SP-RaNN preserves the intrinsic mathematical structure of the equations within a unified space-time framework, ensuring both stability and accuracy. Numerical experiments on the Navier-Stokes, Maxwell, and MHD equations demonstrate that SP-RaNN achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints compared with both traditional numerical methods and DNN-based approaches. This structure-preserving framework provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations

TL;DR

A Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions, and provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

Abstract

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional numerical solvers. To overcome these difficulties, we propose a Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions. Unlike deep neural network (DNN) approaches that rely on expensive nonlinear and nonconvex optimization, SP-RaNN reformulates the training process into a linear least-squares system, thereby eliminating nonconvex optimization. The method linearizes the governing equations through Picard or Newton iterations, discretizes them at collocation points within the domain and on the boundaries using finite-difference schemes, and solves the resulting linear system via a linear least-squares procedure. By design, SP-RaNN preserves the intrinsic mathematical structure of the equations within a unified space-time framework, ensuring both stability and accuracy. Numerical experiments on the Navier-Stokes, Maxwell, and MHD equations demonstrate that SP-RaNN achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints compared with both traditional numerical methods and DNN-based approaches. This structure-preserving framework provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.
Paper Structure (8 sections, 1 theorem, 39 equations, 12 figures, 15 tables)

This paper contains 8 sections, 1 theorem, 39 equations, 12 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Incompressible Magnetohydrodynamics Equations
  3. Structure-Preserving Randomized Neural Networks Method
  4. Randomized Neural Networks
  5. Structure-Preserving RaNNs
  6. SP-RaNN for MHD Equations
  7. Numerical Examples
  8. Summary

Key Result

Lemma 3.1

In a simply connected region, for a vector-valued function $\mathbf{u}:\mathbb{R}^3 \rightarrow\mathbb{R}^3$, the following holds:

Figures (12)

  • Figure 1: The structure of two-hidden-layers neural networks: the dotted line representing weights and bias which are tunable, and the solid line representing weights and bias which are randomly given and fixed.
  • Figure 2: The structure of a two-hidden-layer neural network: the dotted lines represent tunable weights and biases, while the solid lines represent weights and biases that are randomly initialized and fixed.
  • Figure 3: Exact solution, approximation solution of SP-RaNN, and absolute errors of $\mathbf{u}=(u,v)^T$, and $p$, where $m=800,\ \gamma=1,\ N_c=5180,\ r_1=1,\ r_2=0.5$ for Re$=1000$ in Example \ref{['ex_2d_ns']}.
  • Figure 4: Exact solutions, approximation solutions and absolute errors of $\mathbf{u}=(u_1,u_2,u_3)^T$ at $t=1$ on the plane $z=0$, where $m=3200,\ \gamma=100,\ N_c=6144,\ r_1=r_2=0.2$, for ${\rm Re}=1$ in Example \ref{['ex_3d_ns']}.
  • Figure 5: The physical setting of lid-driven cavity flow and activation function in Example \ref{['ex_ben_ns']}.
  • ...and 7 more figures

Theorems & Definitions (14)

  • Lemma 3.1: Pozrikidis1997
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Example 5.1: Steady Two-dimensional Stokes Equations
  • Example 5.2: Unsteady Three-dimensional Beltrami Flow
  • Example 5.3: Lid-driven Cavity Flow
  • ...and 4 more