| Abstract: |
| The incompressible magnetohydrodynamic (MHD) equations are extensively utilized in scientific and engineering fields, yet their strong nonlinearity and two divergence-free conditions pose significant challenges for conventional numerical methods. In this study, we introduce an automatically and precisely divergence-free approach based on Randomized Neural Networks. This method avoids solving nonconvex and nonlinear optimization problems during training, maintains divergence-free properties naturally, and operates as a space-time method. Our proposed approach, named divergence-free randomized neural networks with finite difference method (DF-RNN-FDM), linearizes equations through Picard or Newton iterations, discretizes the problem into a linear system at randomly sampled points across the domain and boundary using the finite difference scheme, and then solves it via a least-square method. We apply this method to solve NS equations, Maxwell equations, and MHD equations. The effectiveness of DF-RNN-FDM is demonstrated by comparison with conventional numerical methods and other neural network-based methods. Our approach achieves higher accuracy with fewer degrees of freedom, simplifies the training process, and precisely adheres to the divergence-free conditions. |
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