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.
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
Figure 1: Illustration of the PML.
Figure 2: Overview of our U-Net architecture, whose main building blocks are described in Tables 1 and 2 .
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
\convcin,cout,k,p,s
Conv2d
in_channels=cin
(k2cin+1)cout
o=⌊(i+2p−k)/s+1⌋
out_channels=cout
kernel_size=k
padding=p
stride=s
\convtcin,cout,k,p,s
ConvTranspose2d
in_channels=cin
(k2cin+1)cout
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.
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.
Figure 4: Evolution of the training and validation loss throughout the training process (reported against the number of gradient steps).
Figure 5: Direct prediction from the neural network on problems with c having different distributions, where the neural network has been trained with α=1 and c∼iidU([1,2]) . Boxplots of ηb based on 1,000 experiments.
Figure 6: Direct prediction from the neural network on problems with α∈{1,2,4,8} , where the neural network has been trained with N=64 and c∼iidU([1,2]) . Boxplots of ηb based on 1,000 experiments.
ϵ
\nnz(A~(c))/\nnz(A(c))
∥A~(c)−A(c)∥F/∥A(c)∥F
0.1
3.411497\cdot10−03 ( 3.060292\cdot10−06 )
1.803282\cdot10−01 ( 6.317130\cdot10−05 )
0.05
6.480183\cdot10−03 ( 2.243749\cdot10−06 )
1.169992\cdot10−01 ( 2.672865\cdot10−05 )
0.02
1.458815\cdot10−02 ( 6.757878\cdot10−06 )
7.725034\cdot10−02 ( 1.337933\cdot10−05 )
0.01
3.222742\cdot10−02 ( 8.740236\cdot10−06 )
3.825805\cdot10−02 ( 9.968938\cdot10−06 )
0.005
4.892559\cdot10−02 ( 8.931253\cdot10−06 )
1.790883\cdot10−02 ( 7.263261\cdot10−06 )
Table 4: Sparsity of A~ in terms of proportion of non-zero (denoted as nnz) entries kept compared to A and relative distance of A~ to A in terms of Frobenius norm, for different values of ϵ . 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}) .
Figure 7: Convergence in terms of ηb 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 α=1 and c∼iidU([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.
Figure 8: Convergence in terms of ηb 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 β , compared with NN-preconditioned FGMRES (NN-FGMRES) corresponding to Training #1, on problems with α=1 and c∼iidU([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.
Figure 9: Convergence in terms of ηb of NN-FGMRES on problems with α=1 and c∼iidU([c0,c1]) with varying ranges [c0,c1] as reported, where the neural network has been trained with α=1 and c∼iidU([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.
Figure 10: Convergence in terms of ηb of NN-FGMRES on problems corresponding to the “square” setting, with different values of c0 and c1 as reported, where the neural network has been trained with α=1 and c∼iidU([1,2]) .
Figure 11: Convergence in terms of ηb of NN-FGMRES on problems with α∈{1,2,4,8} , where the neural network has been trained with α=1 and c∼iidU([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.
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.
Jiwei Jia, Xinliang Liu, Juntao Wang +1
School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China +1
Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.
Mohammad Mahdi Abedi, David Pardo, Tariq Alkhalifah
University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1
This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM-generated training data.