Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations
This provides an efficient tool for scientific and engineering applications involving complex PDEs, but it is incremental as it builds on existing neural network and numerical methods.
The authors tackled the challenge of solving incompressible magnetohydrodynamics equations, which are difficult due to nonlinearity and divergence-free constraints, by proposing a Structure-Preserving Randomized Neural Network that automatically satisfies these constraints and reformulates training into a linear least-squares system, achieving higher accuracy and faster convergence compared to traditional and DNN-based methods.
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.