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The paper trains a U-Net to approximate the inverse of a discretized 2D Helmholtz operator, using a residual-norm loss and no reference solutions, then uses it as a nonlinear preconditioner for FGMRES. The authors are candid that the network is not accurate enough to solve the system alone, but it speeds convergence and, they report, beats LU-based algebraic preconditioners under comparable conditions. Readers should weigh the generalization claims across velocity fields and domain sizes, which are tested on specific parametric families, and treat the broader "extends to other parametric PDEs" claim as untested here.
Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation
arXiv:2610.16542v1 Announce Type: cross Abstract: 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 improves the convergence of FGMRES when employed as a preconditioner. The neural preconditioner demonstrates robust performance and generalization capabilities with respect to variations in the velocity field and the domain size. Comparisons with algebraic preconditioners, based on LU factorizations of approximations of the original system matrices, indicate the superior efficiency of the neural approach under equivalent conditions. We believe that the proposed method is not tied to a specific neural architecture and can be extended to other parametric PDEs.
Abstract: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 improves the convergence of FGMRES when employed as a preconditioner. The neural preconditioner demonstrates robust performance and generalization capabilities with respect to variations in the velocity field and the domain size. Comparisons with algebraic preconditioners, based on LU factorizations of approximations of the original system matrices, indicate the superior efficiency of the neural approach under equivalent conditions. We believe that the proposed method is not tied to a specific neural architecture and can be extended to other parametric PDEs.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2610.16542 [math.NA]
(or arXiv:2610.16542v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2610.16542 arXiv-issued DOI via DataCite (pending registration)
L. Giraud, C. Kruse, P. Mycek, M. Shpakovych, and Y. Xiang, "Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation," Numerical Linear Algebra with Applications 33, no. 5 (2026): e70123
https://doi.org/10.1002/nla.70123 DOI(s) linking to related resources
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