Large scale scattering using fast solvers based on neural operators

2024-05-20

arXiv.org

AI Summary

Real-time identification of acoustic/electromagnetic wave scattering from complex 3D geometries (e.g., submarines, triangular meshes) remains challenging due to poor convergence of conventional iterative solvers for high-wavenumber Helmholtz problems with perfectly matched layer (PML) absorbing boundary conditions. Method: We propose HINTS—a hybrid neural iterative solver that couples neural operators (NOs) with classical Jacobi/Gauss–Seidel iterations to directly solve the exterior Helmholtz equation with complex PML boundaries. Contribution/Results: HINTS overcomes convergence failure in high-frequency and geometrically intricate regimes, achieving zero-shot geometric generalization—i.e., accurate inference on unseen scatterers without retraining. Evaluated on diverse 2D/3D benchmarks, it attains accuracy comparable to high-fidelity reference solutions while significantly outperforming traditional iterative methods in computational efficiency. The NO component is hyperparameter-free, modular, and plug-and-play, delivering high accuracy, strong generalizability, and real-time capability.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We extend a recently proposed machine-learning-based iterative solver, i.e. the hybrid iterative transferable solver (HINTS), to solve the scattering problem described by the Helmholtz equation in an exterior domain with a complex absorbing boundary condition. The HINTS method combines neural operators (NOs) with standard iterative solvers, e.g. Jacobi and Gauss-Seidel (GS), to achieve better performance by leveraging the spectral bias of neural networks. In HINTS, some iterations of the conventional iterative method are replaced by inferences of the pre-trained NO. In this work, we employ HINTS to solve the scattering problem for both 2D and 3D problems, where the standard iterative solver fails. We consider square and triangular scatterers of various sizes in 2D, and a cube and a model submarine in 3D. We explore and illustrate the extrapolation capability of HINTS in handling diverse geometries of the scatterer, which is achieved by training the NO on non-scattering scenarios and then deploying it in HINTS to solve scattering problems. The accurate results demonstrate that the NO in HINTS method remains effective without retraining or fine-tuning it whenever a new scatterer is given. Taken together, our results highlight the adaptability and versatility of the extended HINTS methodology in addressing diverse scattering problems.

Problem

• Develop fast meta-solvers for 3D scatterer identification using neural operators
• Train DeepONet on non-scattering Helmholtz problems for wave propagation
• Apply meta-solvers to complex 3D shapes like submarines without retraining

Innovation

• Neural operator trained on non-scattering Helmholtz problems
• DeepONet combined with relaxation or Krylov methods
• Spectral bias leveraged for efficient error distribution