ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains

Winovich, Nick; Ramani, Karthik; Lin, Guang

doi:10.1016/j.jcp.2019.05.026

Cited by 93 publications

(47 citation statements)

References 33 publications

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“…In summary, the essential idea adopted here is to extract features and learn the corresponding spatial derivatives by using U-net. Except for the very recent work [9], U-net is mainly used for image segmentation, while the machine learning of spatial derivatives is equal to regression tasks (as shown in figure 6). Compared to classical U-net architectures [36], a couple of extensions are performed in this work to achieve effective spatial derivatives, which are summarized as follows:…”

Section: (B) For Problem IImentioning

confidence: 99%

“…The first network model can find practical applications in (radial) wave mode detections, which is a popular topic in the aerospace research community for low-noise aircraft engine design [1][2][3][4], because of the connection between duct acoustics and aeroengine fan noise problems [5][6][7]. The second network can find applications in solving various partial differential equations that has attracted increasing research attentions recently [8][9][10]. The related machine learning details through high-quality training data from the Wiener-Hopf technique are described in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Deep neural networks for waves assisted by the Wiener–Hopf method

Huang

2020

Proc. R. Soc. A.

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In this work, the classical Wiener–Hopf method is incorporated into the emerging deep neural networks for the study of certain wave problems. The essential idea is to use the first-principle-based analytical method to efficiently produce a large volume of datasets that would supervise the learning of data-hungry deep neural networks, and to further explain the working mechanisms on underneath. To demonstrate such a combinational research strategy, a deep feed-forward network is first used to approximate the forward propagation model of a duct acoustic problem, which can find important aerospace applications in aeroengine noise tests. Next, a convolutional type U-net is developed to learn spatial derivatives in wave equations, which could help to promote computational paradigm in mathematical physics and engineering applications. A couple of extensions of the U-net architecture are proposed to further impose possible physical constraints. Finally, after giving the implementation details, the performance of the neural networks are studied by comparing with analytical solutions from the Wiener–Hopf method. Overall, the Wiener–Hopf method is used here from a totally new perspective and such a combinational research strategy shall represent the key achievement of this work.

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Section: (B) For Problem IImentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deep neural networks for waves assisted by the Wiener–Hopf method

Huang

2020

Proc. R. Soc. A.

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“…Thus, recently, ANNs have shown great promise in solving many challenging problems in applied mathematics and computing. In this sense, several ways to solve partial differential equations (PDEs) using feedforward neural networks by substituting approximate solutions into the corresponding differential operator [22,38] have been proposed. For example, Abdeljaber et al [1] studied an interesting active vibration problem of cantilevered beam induced by a pulse concentrated load using an ANN architecture.…”

Section: Introductionmentioning

confidence: 99%

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

2021

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Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

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“…Numerous studies are proposed to solve partial differential equations and have shown great performance in various applications. For example, learning the evolution operator of the PDE from data [27], approximating important physical quantities of the PDE [12], solving heterogeneous elliptic problems on varied domains [26], designing efficient algorithms to handle multiscale multiphase flow problems [25] and using the idea of multiscale model reductions for learning [3,23,24].…”

Section: Introductionmentioning

confidence: 99%