# Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations

arXiv: https://arxiv.org/abs/2603.01102 · HTML: https://mlsanity.com/paper/2603.01102/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 58 references that are in the map.
- First posted on arXiv 1 Mar 2026.

## Builds on (most cited first, 4 of 4)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)

## Built on by (most cited first, 1 of 1)

- [Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems](https://mlsanity.com/paper/2606.19912/) (2026)

## Abstract (arXiv)

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.