Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation

Luc Giraud; Carola Kruse; Paul Mycek; Maksym Shpakovych; Yanfei Xiang

math.NAOct 12, 2026

Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation

Authors: Luc Giraud, Carola Kruse, Paul Mycek, Maksym Shpakovych, Yanfei Xiang

Organizations: Inria, Concace joint team between Airbus CR&T, Cerfacs and Inria · Cerfacs, Concace joint team between Airbus CR&T, Cerfacs and Inria · Inria, Concace joint team between Airbus CR&T, Cerfacs and Inria. · Cerfacs, Concace joint team between Airbus CR&T, Cerfacs and Inria. · Current affiliation: Opticalp · Current affiliation: Opticalp, France. · Current affiliation: IRMA, Université de Strasbourg, MACARON, Inria · Current affiliation: IRMA, Université de Strasbourg, MACARON, Inria, France.

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.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of the PML.

Panel of Figure 2 — Figure 2: Overview of our U-Net architecture, whose main building blocks are described in Tables 1 and 2 .

Panel of Table 1 — Table 1: Description of the modules used in the U-Net architecture depicted in Figure 2 . Each module is defined as a sequence of layers detailed in Table 2 , and the associated number of parameters is deduced from the values provided therein.

Layer	PyTorch name	Arguments	#params	Output size
$\conv_{c_{\textnormal{in}},c_{\textnormal{out}},k,p,s}$	Conv2d	$\texttt{in\_channels}=c_{\textnormal{in}}$	$(k^{2}c_{\textnormal{in}}+1)c_{\textnormal{out}}$	$o=\lfloor(i+2p-k)/s+1\rfloor$
		$\texttt{out\_channels}=c_{\textnormal{out}}$
		$\texttt{kernel\_size}=k$
		$\texttt{padding}=p$
		$\texttt{stride}=s$
$\convt_{c_{\textnormal{in}},c_{\textnormal{out}},k,p,s}$	ConvTranspose2d	$\texttt{in\_channels}=c_{\textnormal{in}}$	$(k^{2}c_{\textnormal{in}}+1)c_{\textnormal{out}}$	$o=s(i-1)-2p+k$

Table 2: Description and PyTorch name of the NN layers used in the U-Net architecture described in Figure 2 and Table 1 , along with their arguments, and the induced number of parameters and output size.

Panel of Figure 3 — Figure 3: Schematic representation of the training framework.

Parameter	Value
Optimizer	Adam [ 12 ]
Mini-batch size $B$	32
Number of epochs	500
Size of data set $K$	$16,000$
Training/validation data set split	96%/4%
Learning rate (LR)	${10}^{-3}$

Table 3: Parameters of the training process.

Panel of Figure 4 — Figure 4: Evolution of the training and validation loss throughout the training process (reported against the number of gradient steps).

Panel of Figure 5 — Figure 5: Direct prediction from the neural network on problems with $c$ having different distributions, where the neural network has been trained with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ . Boxplots of $\eta_{b}$ based on $1,000$ experiments.

Panel of Figure 6 — Figure 6: Direct prediction from the neural network on problems with $\alpha\in\{1,2,4,8\}$ , where the neural network has been trained with $N=64$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ . Boxplots of $\eta_{b}$ based on $1,000$ experiments.

$\epsilon$	$\nnz(\tilde{A}(c))/\nnz(A(c))$	$\\|\tilde{A}(c)-A(c)\\|_{\textnormal{F}}/\\|A(c)\\|_{\textnormal{F}}$
$0.1$	$3.411497\text{\cdot}{10}^{-03}$ ( $3.060292\text{\cdot}{10}^{-06}$ )	$1.803282\text{\cdot}{10}^{-01}$ ( $6.317130\text{\cdot}{10}^{-05}$ )
$0.05$	$6.480183\text{\cdot}{10}^{-03}$ ( $2.243749\text{\cdot}{10}^{-06}$ )	$1.169992\text{\cdot}{10}^{-01}$ ( $2.672865\text{\cdot}{10}^{-05}$ )
$0.02$	$1.458815\text{\cdot}{10}^{-02}$ ( $6.757878\text{\cdot}{10}^{-06}$ )	$7.725034\text{\cdot}{10}^{-02}$ ( $1.337933\text{\cdot}{10}^{-05}$ )
$0.01$	$3.222742\text{\cdot}{10}^{-02}$ ( $8.740236\text{\cdot}{10}^{-06}$ )	$3.825805\text{\cdot}{10}^{-02}$ ( $9.968938\text{\cdot}{10}^{-06}$ )
$0.005$	$4.892559\text{\cdot}{10}^{-02}$ ( $8.931253\text{\cdot}{10}^{-06}$ )	$1.790883\text{\cdot}{10}^{-02}$ ( $7.263261\text{\cdot}{10}^{-06}$ )

Table 4: Sparsity of

\tilde{A}

in terms of proportion of non-zero (denoted as nnz) entries kept compared to

A

and relative distance of

\tilde{A}

A

in terms of Frobenius norm, for different values of

\epsilon

. The average value over 100 experiments is reported, along with the standard deviation between parentheses. Note that

\nnz(A(c))=

7,547,926$$ , and

\|A(c)\|_{F}=

1.110106\text{\cdot}{10}^{03}

\penalty\ (

1.833749\text{\cdot}{10}^{-01}

)

Panel of Figure 7 — Figure 7: Convergence in terms of $\eta_{b}$ of unpreconditioned GMRES, preconditioned GMRES (PGMRES) with the algebraic preconditioners described in Table 4 and FGMRES preconditioned with the trained neural network (NN-FGMRES) corresponding to Training #1, on problems with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ (same as for the neural network training). The iteration-wise median of the convergence history over 100 experiments is displayed as a thick line, while the individual 100 realizations are plotted as thinner, lighter lines.

Panel of Figure 8 — Figure 8: Convergence in terms of $\eta_{b}$ of the system-dependent (Fig. 8(a) , CSL-PGMRES) and average-based (Fig. 8(b) , avg-CSL-PGMRE) CSL-preconditioned GMRES with various values of $\beta$ , compared with NN-preconditioned FGMRES (NN-FGMRES) corresponding to Training #1, on problems with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ (same as for the neural network training). The iteration-wise median of the convergence history over 100 experiments is displayed as a thick line, while the individual 100 realizations are plotted as thinner, lighter lines.

Panel of Figure 9 — Figure 9: Convergence in terms of $\eta_{b}$ of NN-FGMRES on problems with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([c_{0},c_{1}])$ with varying ranges $[c_{0},c_{1}]$ as reported, where the neural network has been trained with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ . The iteration-wise median of the convergence history over 100 experiments is displayed as a thick line, while the individual 100 realizations are plotted as thinner, lighter lines.

Panel of Figure 10 — Figure 10: Convergence in terms of $\eta_{b}$ of NN-FGMRES on problems corresponding to the “square” setting, with different values of $c_{0}$ and $c_{1}$ as reported, where the neural network has been trained with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ .

Panel of Figure 11 — Figure 11: Convergence in terms of $\eta_{b}$ of NN-FGMRES on problems with $\alpha\in\{1,2,4,8\}$ , where the neural network has been trained with $\alpha=1$ and $c\stackrel{{\scriptstyle\textnormal{iid}}}{{\sim}}\mathcal{U}([1,2])$ . The iteration-wise median of the convergence history over 100 experiments is displayed as a thick line, while the individual 100 realizations are plotted as thinner, lighter lines.

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Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Abstract

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