Solving for high-dimensional committor functions using artificial neural networks

Khoo, Yuehaw; Lu, Jianfeng; Ying, Lexing

doi:10.1007/s40687-018-0160-2

Cited by 130 publications

(95 citation statements)

References 20 publications

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“…Another appealing approach that exploits the variational form of PDEs is considered in [9,10,11]. In [9], a committer function is parameterized by a neural network whose weights are obtained by optimizing the variational formulation of the corresponding PDE. In [10], deep learning technique is employed to solve low-dimensional random PDEs based on both strong form and variational form.…”

Section: Related Workmentioning

confidence: 99%

Weak adversarial networks for high-dimensional partial differential equations

Zang

Bao

et al. 2020

Journal of Computational Physics

268

171

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Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.

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Section: Related Workmentioning

confidence: 99%

Weak adversarial networks for high-dimensional partial differential equations

Zang

Bao

et al. 2020

Journal of Computational Physics

268

171

View full text Add to dashboard Cite

show abstract

“…Based on the way that the NN is used, these methods for solving the PDE can be roughly separated into two different categories. For the methods in the first category [16,27,3,10,14,6], instead of specifying the solution space via the choice of basis (as in finite element method or Fourier spectral method), NN is used for representing the solution. Then an optimization problem, for example an variational formulation, is solved in order to obtain the parameters of the NN and hence the solution to the PDE.…”

Section: Introductionmentioning

confidence: 99%

SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

Khoo¹,

Ying²

2019

SIAM J. Sci. Comput.

Self Cite

106

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We propose a novel neural network architecture, SwitchNet, for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered field (and vice versa). The main difficulty of using a neural network for this problem is that a scatterer has a global impact on the scattered wave field, rendering typical convolutional neural network with local connections inapplicable. While it is possible to deal with such a problem using a fully connected network, the number of parameters grows quadratically with the size of the input and output data. By leveraging the inherent low-rank structure of the scattering problems and introducing a novel switching layer with sparse connections, the SwitchNet architecture uses much fewer parameters and facilitates the training process. Numerical experiments show promising accuracy in learning the forward and inverse maps between the scatterers and the scattered wave field.

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“…Problem statement. 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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Solving for high-dimensional committor functions using artificial neural networks

Cited by 130 publications

References 20 publications

Weak adversarial networks for high-dimensional partial differential equations

Weak adversarial networks for high-dimensional partial differential equations

SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

Meta-learning pseudo-differential operators with deep neural networks

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