BCR-Net: A neural network based on the nonstandard wavelet form

Fan, Yuwei; Bohorquez, Cindy Orozco; Ying, Lexing

doi:10.1016/j.jcp.2019.02.002

Cited by 47 publications

(39 citation statements)

References 41 publications

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“…Based on H-matrix and H 2 -matrix structure, Fan and his coauthors proposed two multiscale neural networks [20,21], which are more suitable in training smooth linear or nonlinear operators due to the multiscale nature of H-matrices. In addition to that, nonstandard wavelet form inspired the design of BCR-Net [22], which is applied to address the inverse of elliptic operator and nonlinear homogenization problem and recently been embedded in a neural network for solving electrical impedance tomography [19] and pseudo-differential operator [23]. Multigrid method also inspired MgNet [25].…”

Section: Related Workmentioning

confidence: 99%

Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Li¹,

Cheng²,

Lu³

2020

CICP

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Section: Related Workmentioning

confidence: 99%

Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Li¹,

Cheng²,

Lu³

2020

CICP

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“…with periodic boundary condition. The second positivity constraint in (12) can be dropped since if upxq is the eigenvector associated to the smallest eigenvalue then so is´upxq. Besides, the righthand side of the first constraint in (12) can take any positive constant since this eigenvalue problem is linear.…”

Section: Linear Schrödinger Equationmentioning

confidence: 99%

“…The second positivity constraint in (12) can be dropped since if upxq is the eigenvector associated to the smallest eigenvalue then so is´upxq. Besides, the righthand side of the first constraint in (12) can take any positive constant since this eigenvalue problem is linear. The external potential is randomly generated to simulate crystal with two different atoms in each unit cell, i.e., V pxq is randomly generated via,…”

Section: Linear Schrödinger Equationmentioning

confidence: 99%

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Variational training of neural network approximations of solution maps for physical models

Mao

2020

Journal of Computational Physics

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A novel solve-training framework is proposed to train neural network in representing low dimensional solution maps of physical models. Solve-training framework uses the neural network as the ansatz of the solution map and train the network variationally via loss functions from the underlying physical models. Solve-training framework avoids expensive data preparation in the traditional supervised training procedure, which prepares labels for input data, and still achieves effective representation of the solution map adapted to the input data distribution. The efficiency of solve-training framework is demonstrated through obtaining solutions maps for linear and nonlinear elliptic equations, and maps from potentials to ground states of linear and nonlinear Schrödinger equations. without explicit expression for the underlying model. Benefiting from such a difference, we design loss functions based on the PDEs, in another word, we adopt the model information into the loss functions, and solve the solution map directly without knowing solution functions. Related workA number of recent work utilized NNs to address physical models. Generally, they can be organized into three groups: representing solutions via NNs, representing solution maps via NNs, and optimizing traditional iterative solvers via NNs. Representing solutions of physical models, especially high-dimensional ones, has been a long-standing computational challenge. NN with multiple input and single output can be used as an ansatz for the solutions of physical models or PDEs, which is first explored in [29] for low-dimensional solutions. Many high-dimensional problems, e.g., interacting spin models, high-dimensional committor functions, etc., have been recently considered for solutions using NN ansatz with variants optimization strategies [5,7,9,19,26,37,36]. NN, in this case, is valuable in its flexibility and richness in representing high-dimensional functions.Representing the solution map of a nonlinear problem is challenging as well. For linear problems, the solution map can be represented by a simple matrix (i.e., Green's function for PDE problems). While the efficient representation for solution map is unknown for most nonlinear problems. Traditional methods in turn solve nonlinear problem via iterative methods, e.g., fixed point iteration. Since NN is able to represent high dimensional nonlinear mappings, it has also been explored in recent literature to represent solution maps of low-dimensional problems on mesh grid, see e.g, [10,11,12,20,25,27,30,34,38,39]. These NNs are fitted by a set of training data with solution ready, i.e., labeled data. Most work from the first two groups focus on creative design of NN architectures, in particular trying to incorporate knowledge of the PDE into the representation.The last group, very different from previous two, adopts NN to optimize traditional iterative methods [14,23,24,35]. Once the iterative methods are optimized on a set of problems, generalization to different boundary conditions, domain geometries, and...

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“…This paper is concerned with a more ambitious task: representing the nonlinear map from η to G η M : η → G η = L −1 η , (1.2) approximate high-dimensional solutions of high-dimensional PDEs [36,52,14,46,4,6,13,25,31,39]. In a somewhat orthogonal direction, the NNs have been utilized to approximate the high-dimensional parameterto-solution of various PDEs and IEs [30,26,18,17,19,32,25,2,38,20].…”

Section: Introductionmentioning

confidence: 99%

Meta-learning pseudo-differential operators with deep neural networks

Feliu‐Fabà

Fan

Ying

2020

Journal of Computational Physics

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This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed form with a collection of vectors. The nonlinear map from the parameter to this collection of vectors and the wavelet transform are learned together from a small number of matrix-vector multiplications of the pseudo-differential operator. Numerical results for Green's functions of elliptic partial differential equations and the radiative transfer equations demonstrate the efficiency and accuracy of the proposed approach.Keywords: Deep neural networks; Convolutional neural networks; Nonstandard wavelet form; Metalearning; Green's functions; Radiative transfer equation.when the operator G η is a pseudo-differential operator (PDO) [57]. Although L η and G η can be linear operators, this map M from η to G η is highly nonlinear.Background. In the recent years, deep learning has become the most versatile and effective tool in artificial intelligence and machine learning, witnessed by impressive achievements in computer vision [35,58,28], speech and natural language processing [29,53,48,54,11], drug discovery [41] or game playing [51,15,55]. Recent reviews on deep learning and its impacts on other fields can be found in for example [37,50]. At the center of deep learning, the model of deep neural networks (NNs) provides a flexible framework for approximating high-dimensional functions, while allowing for efficient training and good generalization properties in practice [40,44].More recently, several groups have started applying NNs to partial differential equations (PDEs) and integral equations (IEs) arising from physical systems. In one direction, the NN model has been used to

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BCR-Net: A neural network based on the nonstandard wavelet form

Cited by 47 publications

References 41 publications

Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks

Variational training of neural network approximations of solution maps for physical models

Meta-learning pseudo-differential operators with deep neural networks

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