[2603.01102] Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations

Physics > Fluid Dynamics arXiv:2603.01102 (physics) [Submitted on 1 Mar 2026] Title:Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations Authors:Yunlong Li, Fei Wang, Lingxiao Li View a PDF of the paper titled Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations, by Yunlong Li and 1 other authors View PDF HTML (experimental) 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 stab...