Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation
This work presents a hybrid numerical approach for solving linear systems arising from the discretization of the two-dimensional parametric Helmholtz equation. A convolutional neural network based on the U-Net architecture is trained in an self-supervised manner (i.e., without explicitly generating reference solutions) to approximate the inverse of the discretized Helmholtz operator, using a loss function involving the residual norm of the linear system. The trained network is used as a nonlinear preconditioner within the flexible GMRES (FGMRES) algorithm. Numerical experiments show that while the neural network is not accurate enough to act as a standalone solver, it significantly improv...
| Authors | Luc Giraud, Carola Kruse, Paul Mycek et al. |
|---|---|
| Primary category | math.NA |
| Published | 12 October 2026 |
| Last updated at source | 12 October 2026 |
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