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Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

Machine Learning · Computer Science 2025-10-20 Ziqian Li , Kang Liu , Yongcun Song , Hangrui Yue , Enrique Zuazua

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated…

Numerical Analysis · Mathematics 2016-12-14 Michael V. Klibanov , Dinh-Liem Nguyen , Loc H. Nguyen , Hui Liu

Metasurfaces have become a promising means for manipulating optical wavefronts in flat and high-performance optical devices. Conventional metasurface device design relies on trial-and-error methods to obtain target electromagnetic (EM)…

Tracking a horizon in seismic images or 3D volumes is an integral part of seismic interpretation. The last few decades saw progress in using neural networks for this task, starting from shallow networks for 1D traces, to deeper…

Geophysics · Physics 2020-03-20 Bas Peters , Eldad Haber

Metasurfaces are extremely useful for controlling and manipulating electromagnetic waves. Full-wave numerical simulation is highly desired for their design and optimization, but it is notoriously difficult, even for two-dimensional…

Optics · Physics 2026-01-21 Fuhao Liu , Ya Yan Lu

We propose a deep learning framework based on an encoder-decoder architecture for the design and evaluation of cloaking devices, demonstrated in this work for two-dimensional wave propagation governed by the Helmholtz equation. The cloaks…

Computational Physics · Physics 2026-05-22 Camille Carvalho , Elsie Cortes , Chrysoula Tsogka , Symeon Papadimitropoulos

The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme}…

Machine Learning · Computer Science 2021-10-12 Gaurav Gupta , Xiongye Xiao , Paul Bogdan

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited…

Analysis of PDEs · Mathematics 2020-09-10 Florian Faucher , Otmar Scherzer , Hélène Barucq

Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs…

Machine Learning · Computer Science 2026-03-17 Yiyang Cai , Zixuan Qiu , Yunlu Shu , Jiamao Wu , Yingzhou Li , Tianyu Wang , Xi Chen

Fourier Neural Operators (FNOs) have been promoted as fast, mesh-invariant surrogates for partial-differential equation solvers, with seismic studies reporting orders-of-magnitude speedup over classical methods. We revisit those claims by…

Geophysics · Physics 2025-08-18 Dimitri Voytan , Litan Li

We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning,…

Machine Learning · Computer Science 2024-06-07 Bar Lerer , Ido Ben-Yair , Eran Treister

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…

Numerical Analysis · Mathematics 2026-02-05 Victorita Dolean , Daria Hrebenshchykova , Stéphane Lanteri , Victor Michel-Dansac

Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these…

Machine Learning · Computer Science 2025-06-27 Peiyan Hu , Rui Wang , Xiang Zheng , Tao Zhang , Haodong Feng , Ruiqi Feng , Long Wei , Yue Wang , Zhi-Ming Ma , Tailin Wu

Deep learning applications are drastically progressing in seismic processing and interpretation tasks. However, the majority of approaches subsample data volumes and restrict model sizes to minimise computational requirements. Subsampling…

Geophysics · Physics 2021-02-26 Claire Birnie , Haithem Jarraya , Fredrik Hansteen

The manuscript presents a new technique for computing the exponential of skew-Hermitian operators. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many…

Numerical Analysis · Mathematics 2014-02-24 T. S. Haut , T. Babb , P. G. Martinsson , B. A. Wingate

From ancient to modern times, acoustic structures have been used to control the propagation of acoustic waves. However, the design of the acoustic structures has remained widely a time-consuming and computational resource-consuming…

Sound · Computer Science 2024-11-12 Xuecong Sun , Han Jia , Yuzhen Yang , Han Zhao , Yafeng Bi , Zhaoyong Sun , Jun Yang

The precise simulation of turbulent flows is of immense importance in a variety of scientific and engineering fields, including climate science, freshwater science, and the development of energy-efficient manufacturing processes. Within the…

Fluid Dynamics · Physics 2024-06-10 Shengyu Chen , Peyman Givi , Can Zheng , Xiaowei Jia

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a…

Numerical Analysis · Mathematics 2024-06-21 Zhiwei Zhao , Changqing Liu , Yingguang Li , Zhibin Chen , Xu Liu

Large scale parameter estimation problems are among some of the most computationally demanding problems in numerical analysis. An academic researcher's domain-specific knowledge often precludes that of software design, which results in…

Mathematical Software · Computer Science 2017-04-05 Curt Da Silva , Felix J. Herrmann

Simulating spatiotemporal turbulence with high fidelity remains a cornerstone challenge in computational fluid dynamics (CFD) due to its intricate multiscale nature and prohibitive computational demands. Traditional approaches typically…

Fluid Dynamics · Physics 2024-07-01 Xiantao Fan , Deepak Akhare , Jian-Xun Wang
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