| Abstract: |
| The incompressible magnetohydrodynamic (MHD) equations are fundamental in modeling many
physical and engineering phenomena. Yet their strong nonlinearity and two divergence-free constraints
make them notoriously difficult to solve with traditional numerical methods. Recently, neural networks
have gained attention for solving PDEs, but deep learning approaches often struggle: they are slow to
train, get trapped in local minima, and only approximate mass conservation. In this talk, I will present
a Structure-Preserving Randomized Neural Network (SP-RaNN) designed to overcome these
challenges. SP-RaNN automatically and exactly enforces divergence-free conditions, avoids costly
nonlinear optimization, and works in a space-time framework that preserves both physical and
mathematical structures. The method linearizes the governing equations, discretizes them at
collocation points, and solves the resulting system efficiently with least-squares techniques. Through
applications to Navier-Stokes, Maxwell, and MHD equations, I will show that SP-RaNN achieves
higher accuracy and efficiency than both traditional solvers and deep neural network methods. |
|