Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation

• URL: http://arxiv.org/abs/2511.15445v1
• Date: Wed, 19 Nov 2025 13:58:32 GMT
• Title: Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation
• Authors: Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri, Victor Michel-Dansac,
• Abstract summary: Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis.<n>Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation.<n>This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.

Related papers

• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• Domain decomposition architectures and Gauss-Newton training for physics-informed neural networks [47.614449195824335]
  Approxing boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process.<n>This difficulty can be partly explained by the spectral bias, that is, the slower convergence of high-frequency components.<n>We combine this localization with the Gauss-Newton method as the gradient to obtain faster convergence than gradient-based schemes such as Adam.<n>Our numerical results indicate that combining localization and Gauss-Newton optimization is promising for neural network-based solvers for partial differential equations.
  arXiv  Detail & Related papers  (2025-10-30T21:45:10Z)
• Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks [0.0]
  Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics.<n>Their success in predicting the acoustic field is limited by the vanishing-gradient problem that occurs when solving the Helmholtz equation.<n>The problem of solving the two-dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method.<n>A trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfiniteconstrained and the theory of R-functions.
  arXiv  Detail & Related papers  (2025-03-26T04:28:49Z)
• Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks [0.24578723416255746]
  We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. A distinct advantage our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations.
  arXiv  Detail & Related papers  (2024-09-20T16:48:55Z)
• Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation [0.0]
  Singularly perturbed problems have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods, require a refined mesh to obtain stable and accurate solutions. We consider the use of Physics-Informed Neural Networks (PINNs) to produce numerical solutions of singularly perturbed problems.
  arXiv  Detail & Related papers  (2024-09-12T00:24:21Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Enhancing Solutions for Complex PDEs: Introducing Complementary Convolution and Equivariant Attention in Fourier Neural Operators [17.91230192726962]
  We propose a novel hierarchical Fourier neural operator along with convolution-residual layers and attention mechanisms to solve complex PDEs. We find that the proposed method achieves superior performance in these PDE benchmarks, especially for equations characterized by rapid coefficient variations.
  arXiv  Detail & Related papers  (2023-11-21T11:04:13Z)
• Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems [11.707981310045742]
  Neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. The efficiency and accuracy of the proposed methods are demonstrated for multi-frequency model problems.
  arXiv  Detail & Related papers  (2023-11-07T06:08:47Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Physics informed neural networks for continuum micromechanics [68.8204255655161]
  Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world $mu$CT-scans.
  arXiv  Detail & Related papers  (2021-10-14T14:05:19Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.