ScaGNN: a Graph Neural Network for Multiple Scattering Simulations

Rémi Marsal; Stéphanie Chaillat; Alexandre Chapoutot

cs.LGSep 29, 2026

ScaGNN: a Graph Neural Network for Multiple Scattering Simulations

Authors: Rémi Marsal, Stéphanie Chaillat, Alexandre Chapoutot

Organizations: U2IS, ENSTA, Institut Polytechnique de Paris · Laboratoire POEMS, CNRS, INRIA, ENSTA, Institut Polytechnique de Paris

Abstract

The boundary element method (BEM) provides an efficient numerical framework for solving multiple scattering problems in unbounded homogeneous domains. By restricting the discretization to the domain boundaries, it substantially reduces computational complexity. The procedure first consists in determining the solution trace on the boundaries of the domain by solving a boundary integral equation. Then, the volumetric solution can be recovered at low computational cost using a boundary integral representation. As the first step of the BEM represents the main computational bottleneck, we present ScaGNN, a learning-based approach designed to approximate the solution trace. It relies on a graph neural network architecture that incorporates a dynamic adaptive edge sampling mechanism for selecting the most relevant interactions to model. Guided by intermediate predictions of expected error and edge length, this mechanism selects, at various stages of the forward pass, the most relevant distant interactions to model. The proposed method is tailored to achieve linear complexity with the number of nodes in the input graph. To train and evaluate our network, we present a benchmark consisting of several datasets with different types of multiple scattering problems. Our experiments show that our approach surpasses existing state-of-the-art learning-based methods on the considered tasks and investigate the generalization capabilities to settings with an increased number of obstacles and out-of-distribution obstacle shapes. github.com/LARIAD/ScaGNN

Figures & tables

Figure 1: Illustration of the resulting field

u

(dashed arrows) from the scattering of an incoming wave

u_{\mathrm{inc}}

(solid arrows) by three obstacles

\Omega_{1}

\Omega_{2}

and

\Omega_{3}

Figure 2: Illustration of ScaGNN architecture with

L=3

hierarchical levels. The node and edge encoders have not been represented for better readability. The graph representation processed by each MP layer is given below the corresponding block. The architecture first consists of a Boundary Block with

N_{\mathrm{b}}

MP layers applied to the Boundary Graph

\mathcal{G}^{0}

, then two Downsampling Blocks. At the lowest resolution,

K

Distant Interaction Blocks are applied. They are composed of an intermediate decoder whose outputs are used by our dynamic adaptive edge sampling to select the edges of the Distant Interaction Graph

\mathcal{G}^{2}_{1}

, …,

\mathcal{G}^{2}_{K}

and

N_{\mathrm{d}}

MP layers. Finally, two Upsampling Blocks and a Boundary Block of

N_{\mathrm{b}}

MP layers are applied to the Boundary Graph

\mathcal{G}^{0}

before the final decoder. Gradient flows through solid arrows, not dashed arrows.

Architecture Type		Method	Number of Parameters	FLOPs	Laplace	Helmholtz Dirichlet		Helmholtz Neumann
GNN	Transformer	Method	Number of Parameters	FLOPs	$\mathrm{Err}_{\mathrm{rel}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$
	✓	PTv3	19.2 M	16.8 G	0.081	0.148	0.141	0.070	0.073
	✓	Transolver	4.6 M	38.4 G	0.183	0.596	0.454	0.070	0.073
	✓	Transolver++	4.2 M	35.6 G	0.195	0.635	0.466	0.103	0.122
✓	✓	Erwin	7.9 M	42.4 G	0.119	0.218	0.175	0.074	0.076
✓		MGN	5.3 M	160.4 G	0.173	0.286	0.181	0.068	0.068

Table 1: Benchmark results on in-distribution test datasets comparing the prediction errors, number of parameters and computational cost (in FLOPs).

Panel of Figure 3 — Figure 3: Estimation errors as a function of the number of obstacles for the Laplace (top), the Helmholtz Dirichlet (middle) and the Helmholtz Neumann (bottom) problems, respectively.

Method	Laplace	Helmholtz Dirichlet		Helmholtz Neumann
Method	$\mathrm{Err}_{\mathrm{rel}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$
PTv3	0.166	0.203	0.165	0.071	0.098
Transolver	0.195	0.437	0.381	0.076	0.091
Transolver++	0.206	0.458	0.390	0.087	0.120
Erwin	0.148	0.211	0.199	0.088	0.116
MGN	0.211	0.214	0.158	0.079	0.081

Table 2: Benchmark results on out-of-distribution obstacle shapes test sets.

Figure 4: Runtime comparison with respect to the number of obstacles for learning-based methods and for the BEM, considering different convergence tolerance thresholds

\mathrm{rtol}

for GMRES. The two curves obtained using our method, Ours (

N_{e}=4

) and Ours (

N_{e}=20

), overlap.

Dimension at the finest level	Dimension expansion factor	$K$	$N_{\mathrm{d}}$	Intermediate Predictions	Edge Sampling	# params	FLOPs	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$
184	1	1	6	✗	Uniform	5.6 M	79.2 G	0.109	0.105
64	2	1	6	✗	Uniform	5.2 M	29.0 G	0.106	0.104
32	3	1	6	✗	Uniform	5.8 M	26.3 G	0.128	0.128
64	2	3	2	✗	Uniform	5.2 M	29.4 G	0 .100	0 .099
64	2	3	2	✓	Uniform	5.6 M	29.6 G	0 .100	0 .099
64	2	3	2	✓	Adaptive	5.6 M	29.6 G	0.087	0.086

Table 3: Ablation

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Illustration of some directed graph representations used in ScaGNN in the case of two obstacles. Graph edges are in dark blue, while the light blue corresponds to the obstacle meshes. $\mathcal{G}^{0}$ : the Boundary Graph (arrows have been omitted for readability), $\mathcal{G}^{1\rightarrow 2}$ : the Downsampling Graph from level 1 to 2, $\mathcal{G}^{2}$ : a Distant Interaction Graph and $\mathcal{G}^{2\rightarrow 1}$ : the Upsampling Graph from level 2 to 1. We omit the Downsampling and Upsampling Graphs $\mathcal{G}^{0\rightarrow 1}$ and $\mathcal{G}^{1\rightarrow 0}$ between levels 0 and 1 which are similar to graphs $\mathcal{G}^{1\rightarrow 2}$ and $\mathcal{G}^{2\rightarrow 1}$ at a lower resolution.

Datasets	Training Set	Test Sets
Datasets	Training Set	3 obstacles	6 obsatcles	9 obstacles	OoD
Number of samples	10k	1k	1k	1k	1k
Number of obstacles per samples	3	3	6	9	3
Total number of nodes	41M	4M	8M	12M	3M
Obstacle shape	Ellipsoid	Ellipsoid	Ellipsoid	Ellipsoid	Rounded parallelepiped

Appendix

Table 4: General dataset statistics that are common for all problems

Environment size	$10\times 10\times 10$
Edge length in obstacle meshes	0.1
Minimal distance between two obstacles	0.1
Ellipses semi-axes length (min – max)	0.3 – 1.5
Wavelength (min – max)	0.6 – 6
Rounded parallelepiped length, width, height (min – max)	0.6 – 3
Rounded parallelepiped rounded radius length (min – max)	0.3 – max(parallelepiped length, width, height)

Appendix

Table 5: Main characteristics of our datasets. Lengths are given without units.

Problem	BIE	Representation
Laplace Dirichlet	$\mathrm{S}p=u$	$u=\mathcal{S}p$
Helmholtz Dirichlet	$\mathrm{S}p=u$	$u=\mathcal{S}p$
Helmholtz Neumann	$\mathrm{N}p=\frac{\partial u}{\partial\mathbf{n}}$	$u=\mathcal{D}p$

Appendix

Table 6: Formulations of the BIE and of the representations used to generate the dataset for each problem where

\mathrm{S}

and

\mathrm{N}

are the single-layer and the hypersingular boundary integral operators, respectively, while

\mathcal{S}

and

\mathcal{D}

are the single-layer and the double-layer potential operators, respectively.

Panel of Figure 6 — Figure 6: Illustrations of dataset samples with obstacle meshes in blue and the source in red.

Encoder	Input Feature	Dimensions
Node encoder	Sinusoidal encoding of the distance between the current node and $\mathbf{x}_{0}$	128
	Normalized direction of $\mathbf{x}_{0}$ relative to the current node	3
	Normalized direction $\mathbf{v}$	3
	Each term of the boundary condition computed at the current node position	3
Edge encoder	Sinusoidal encoding of the edge length	128
Edge encoder	Normalized direction of the edge	3

Appendix

Table 7: Inputs of the node and edge encoders, and their corresponding dimensionalities for the Laplace Dirichlet problems.

Encoder	Input Feature	Dimensions
Node encoder	Sinusoidal encoding of the distance between the current node and the source $\mathbf{x}_{0}$	128
	Normalized direction of the source $\mathbf{x}_{0}$ relative to the current node	3
	Wavenumber $k$ of the sample	1
	Sine and cosine of the angle of the incident wave with wavenumber $k$ coming from $\mathbf{x}_{0}$	2
Edge encoder	Sinusoidal encoding of the edge length	128
	Normalized direction of the edge	3

Appendix

Table 8: Inputs of the node and edge encoders, and their corresponding dimensionalities for the Helmholtz Dirichlet problems.

Encoder	Input Feature	Dimensions
Node encoder	Normalized direction of $\mathbf{v}$ of the incident wave	3
	Wavenumber $k$ of the sample	1
	Sine and cosine of the angle of the incident wave with wavenumber $k$	2
	Sinusoidal encoding of the distance between the current node and the average of node positions	128
	Normalized direction of the average node position	3
Edge encoder	Sinusoidal encoding of the edge length	128

Appendix

Table 9: Inputs of the node and edge encoders, and their corresponding dimensionalities for the Helmholtz Neumann problems.

Model	Parameter	Value
MeshGraphNet ( Pfaff et al., 2020 )	Processor depth	15
	Latent dimension	128
Point Transformer V3 ( Wu et al., 2024b )	Grid Size	0.2
	Encoder Latent dimensions	(64, 128, 256)
	Encoder depths	(2, 2, 6)
	Encoder heads	(4, 8, 16)

Appendix

Table 10: Implementation details of the different methods we compare against in this paper.

Method	Number of obstacles	Laplace	Helmholtz Dirichlet		Helmholtz Neumann
Method	Number of obstacles	$\mathrm{Err}_{\mathrm{rel}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$	$\mathrm{Err}_{\mathrm{ampl}}$	$\mathrm{Err}_{\mathrm{angle}}$
Ours ( $N_{\mathrm{e}}=4$ )	3	0.29%	0.20%	0.21%	0.08%	0.12%
Ours ( $N_{\mathrm{e}}=4$ )	6	0.14%	0.39%	0.11%	0.07%	0.08%
Ours ( $N_{\mathrm{e}}=4$ )	9	0.18%	0.20%	0.20%	0.04%	0.03%
Ours ( $N_{\mathrm{e}}=20$ )	3	0.31%	0.18%	0.21%	0.10%	0.09%
Ours ( $N_{\mathrm{e}}=20$ )	6	0.24%	0.15%	0.15%	0.13%	0.09%

Appendix

Table 11: Relative standard deviation results over five inferences.

Figure 7: Computational cost of ScaGNN with

N_{\mathrm{e}}=4

(in blue) and

N_{\mathrm{e}}=20

(in orange) as a function of the number of nodes per sample in the test sets with 3, 6 and 9 obstacles.

Index of the Distant Interaction Block	1	2	3
Inverse distance to the source	0.65	0.46	0.50
Inverse distance to the closest obstacle	0.21	0.51	0.47

Appendix

Table 12: Correlations between the expected error predicted by each Distant Interaction Block and the inverse of the two distances: the source distance and the distance to the closest obstacle.

Panel of Figure 8 — Figure 8: Expected error predicted by ScaGNN intermediate decoders for a sample from the Helmholtz Dirichlet test set. It is worth noting that the intermediate decoder predictions on the low-resolution graph have been interpolated to the original mesh.

	Laplace	Helmholtz Dirichlet	Helmholtz Neumann
log number of GMRES iterations	-0.15	0.80	0.76
Log wavenumber	–	0.81	0.81
Obstacle dispersion	0.06	-0.23	-0.11

Appendix

Table 13: Correlations between the log mean absolute error and several dataset sample features.

Panel of Figure 9 — Figure 9: Estimation errors as a function of the number of edges $N_{\mathrm{e}}$ per Distant Interaction Graph.“All” means that the Distant Interaction Graph is densely connected.

Panel of Figure 10 — Figure 10: Estimation errors as a function of the number $K$ of distinct Distant Interaction Graphs processed in our GNN to model distant interactions.

K	Number of parameters	FLOPs
1	5.4 M	29.1 G
2	5.5 M	29.3 G
3	5.6 M	29.6 G
6	6.0 M	30.4 G

Appendix

Table 14: Number of parameters and FLOPs for different values of K.

Panel of Figure 11 — Figure 11: Estimation errors as a function of the $\alpha$ parameter in the score function $f_{\mathrm{score}}^{\alpha}$ . When $\alpha=\infty$ , the score function only depends on error predictions and the edge length is ignored.

Panel of Figure 12 — Figure 12: Estimation errors with three obstacles as a function of the $\gamma$ parameter that weights the intermediate predictions in the loss function $\mathcal{L}_{\mathrm{total}}$ .

Panel of Figure 13 — Figure 13: Estimation errors with three obstacles as a function of the number $C$ of candidate edges from which each edge in a Distant Interaction Graph $\mathcal{G}^{L-1}_{k}$ , $1\leq k\leq K$ , is selected. More specifically, $C$ varies during both training and testing.

Panel of Figure 14 — Figure 14: Estimation errors as a function of the number $C$ of candidate edges for selecting each edge in the Distant Interaction Graph $\mathcal{G}^{L-1}_{k}$ , $1\leq k\leq K$ . Here, $C$ varies at test time while the evaluated model has been trained with $C=2$ .

Panel of Figure 15 — Figure 15: Qualitative results on the Laplace Dirichlet problem with Point Transformer v3 ( Wu et al., 2024b ) and ScaGNN.

Panel of Figure 16 — Figure 16: Qualitative results for amplitude predictions on the Helmholtz Dirichlet problem with Point Transformer v3 ( Wu et al., 2024b ) and ScaGNN.

Panel of Figure 17 — Figure 17: Qualitative results for angle predictions on the Helmholtz Dirichlet problem with Point Transformer v3 ( Wu et al., 2024b ) and ScaGNN.

Panel of Figure 18 — Figure 18: Qualitative results for amplitude predictions on the Helmholtz Neumann problem with Point Transformer v3 ( Wu et al., 2024b ) and ScaGNN.

Panel of Figure 19 — Figure 19: Qualitative results for angle predictions on the Helmholtz Neumann problem with Point Transformer v3 ( Wu et al., 2024b ) and ScaGNN.

Figure 20: From left to right are the volumetric solutions of the total field obtained with GMRES (the groundtruth), with ScaGNN and with PTv3 for estimating the trace solution on the boundary, respectively, and the corresponding errors relative to the groundtruth with ScaGNN and PTv3, respectively. For each problem, the volumetric solutions of the total field and their associated errors are sampled within a square domain of side length 10 on the plane $z=0$ , and the obstacles are represented in white.

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An Attention-Guided Graph Neural Network Surrogate for Boundary Integral Computation in Physical Field Reconstruction

RongZe Gao, YuWei Wu, XinRan Ji, XingYue Ren

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PDE Surrogate Modeling PDE Inverse Problems PDE Solving Graph Neural Networks

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In practical engineering and scientific computing, a real physical system can be reduced to a boundary value problem, and the adequate boundary information to determine the interior spatial distribution and evolutionary state of the physical field can be solved with linear equations, alternatively, the the real physical problem can be treated with boundary problems with in adequate information problems in field of machine learning. This paper proposes a deep learning framework for the recovery of missing boundary information using an attention-guided graph neural network (GNN) surrogate. The framework extracts features from the physical information on the discretized boundary into a latent space, and the missing boundary quantities are then predicted by the network. Thus, the nonlinear mapping from known boundary quantities to missing boundary quantities is learned using a variational autoencoder (VAE) architecture, while an attention mechanism is incorporated to better model the correlations among different types of boundary information. Furthermore, the interior physical field can be reconstructed using the conventional boundary element method (BEM) integral formulation. The effectiveness of the GNN-BEM framework is evaluated using numerical examples of the two-dimensional Laplace and Poisson equations, which demonstrate good performance in reconstruction accuracy, local detail preservation, and model stability. The results show that GNN-BEM provides a feasible computational framework for boundary information recovery and fast interior physical field reconstruction for partial differential equations under complex boundary conditions.

Abstract

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