Learned Iterative Solvers for the Helmholtz Equation

Rizzuti, Gabrio; Siahkoohi, Ali; Herrmann, Felix J.

doi:10.3997/2214-4609.201901542

Cited by 18 publications

(15 citation statements)

References 10 publications

(12 reference statements)

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“…A key part of their approach is ensuring that the learned scheme has convergence guarantees. See also Rizzuti, Siahkoohi and Herrmann (2019) for a similar approach to solving the Helmholtz equation.…”

Section: Learning In Functional Analytic Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

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Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

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Section: Learning In Functional Analytic Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

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“…In [27], the authors propose an iterative solver for the Helmholtz equation which combines traditional Krylovbased solvers with machine learning. The result is a reduced computational complexity.…”

Section: Introductionmentioning

confidence: 99%

A Feedforward Neural Network for Modeling of Average Pressure Frequency Response

2022

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The Helmholtz equation has been used for modeling the sound pressure field under a harmonic load. Computing harmonic sound pressure fields by means of solving Helmholtz equation can quickly become unfeasible if one wants to study many different geometries for ranges of frequencies. We propose a machine learning approach, namely a feedforward dense neural network, for computing the average sound pressure over a frequency range. The data are generated with finite elements, by numerically computing the response of the average sound pressure, by an eigenmode decomposition of the pressure. We analyze the accuracy of the approximation and determine how much training data is needed in order to reach a certain accuracy in the predictions of the average pressure response.

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“…Thus, we explore the machine learning (ML) world for a more practical solution. Recently, Rizzuti et al (2019) built on the ML ability to predict the sequence evolution of iterative solvers in representing the Krylov-based iterative solver for the Helmholtz equation. For a large model size, their stiffness matrix based approach can be costly.…”

Section: Introductionmentioning

confidence: 99%

Wavefield Solutions from Machine Learned Functions that Approximately Satisfy the Wave Equation

Alkhalifah¹,

Song²,

Waheed³

et al. 2020

EAGE 2020 Annual Conference &Amp; Exhibition Online

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Solving the Helmholtz equation provides wavefield solutions that are dimensionally compressed, per frequency, compared to the time domain which is useful for many applications, like full waveform inversion (FWI). However, the efficiency in attaining such wavefield solutions depends often on the size of the model, which tends to be large at high frequencies and for 3D problems. Thus, we use here a recently introduced framework based on neural networks to predict such solutions through setting the underlying physical equation as a loss function to optimize the neural network parameters. For an input point in the model space, the network learns to predict the wavefield value at that point, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions. Feeding a reasonable number of random points from the model space will ultimately train a fully connected 8-layer deep neural network with each layer having a dimension of 20, to predict the scattered wavefield function. Initial tests on a two-box-shaped scatterer model with a source in the middle, as well as, a layered model with a source on the surface demonstrate the successful training of the NN for this application and provide us with a peek into the potential of such an approach.

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Learned Iterative Solvers for the Helmholtz Equation

Cited by 18 publications

References 10 publications

Solving inverse problems using data-driven models

Solving inverse problems using data-driven models

A Feedforward Neural Network for Modeling of Average Pressure Frequency Response

Wavefield Solutions from Machine Learned Functions that Approximately Satisfy the Wave Equation

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