A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems

Hu, Wen‐Chen; Lin, Te-Sheng; Lai, Ming-Chih

doi:10.48550/arxiv.2106.05587

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Moreover, we can extend this to any function which has a finite number of discontinuities. This is done using the discontinuity capturing shallow neural network (DCSNN) technique described in [11]. The key idea is that we can extend a d-dimensional piecewise continuous function to a (d + 1)-dimensional continuous function by adding a new variable to parameterize the domain.…”

Section: Universal Approximation Theoremmentioning

confidence: 99%

“…As discussed earlier in the paper, the DCSNN technique described in [11] is utilized to deal with the discontinuous constant coefficients across the two domains. Using the protocol described in equations 11 and 12, an extra feature is introduced to extend the discontinuous function in d dimensions to a continuous function in d + 1 dimensions.…”

Section: Algorithm Formulationmentioning

confidence: 99%

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Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Mills¹,

Pozdnyakov²

2022

Preprint

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Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex differential equations and necessitating sophisticated numerical methods to approximate solutions. Trained neural networks act as universal function approximators, able to numerically solve differential equations in a novel way. In this work, methods and applications of neural network algorithms for numerically solving differential equations are explored, with an emphasis on varying loss functions and biological applications. Variations on traditional loss function and training parameters show promise in making neural network-aided solutions more efficient, allowing for the investigation of more complex equations governing biological principles.

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Section: Universal Approximation Theoremmentioning

confidence: 99%

Section: Algorithm Formulationmentioning

confidence: 99%

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Mills¹,

Pozdnyakov²

2022

Preprint

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“…However, the above immersed interface method becomes more difficult and tedious to implement when the space dimension increases. Recently, the authors of this paper propose a machine learning method called discontinuity capturing shallow neural network (DCSNN) [17] to solve more general elliptic interface problems than the one in Eq. ( 3) and successfully obtain the comparable results with IIM.…”

Section: Elliptic Problems With Singular Sources On the Interfacementioning

confidence: 99%

“…We note that the construction of such a set of functions using neural net approximation requires additional efforts. Here, inspired by DCSNN [17] where an augmented variable is introduced to categorize precisely the variables into each subdomains, we also introduce an augmented variable and require the function to be continuous throughout the whole domain. More precisely, consider a level set function φ(x) such that the position of the interface Γ is given by the zero level set, i.e.,…”

Section: Level Set Function Augmentationmentioning

confidence: 99%

A Shallow Ritz Method for Elliptic Problems with Singular Sources

Lai,

Chang,

Lin

et al. 2021

Preprint

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In this paper, a shallow Ritz-type neural network for solving elliptic problems with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function singularity is naturally removed, (ii) level set function is introduced as a feather input, (iii) it is completely shallow consisting of only one hidden layer. We first introduce the energy functional of the problem and then transform the contribution of singular sources to a regular surface integral along the interface. In such a way the delta function singularity can be naturally removed without the introduction of discrete delta function that is commonly used in traditional regularization methods such as the well-known immersed boundary method. The original problem is then reformulated as a minimization problem. We propose a shallow Ritz-type neural network with one hidden layer to approximate the global minimizer of the energy functional. As a result, the network is trained by minimizing the loss function that is a discrete version of the energy. In addition, we include the level set function of the interface as a feature input and find that it significantly improves the training efficiency and accuracy. We perform a series of numerical tests to demonstrate the accuracy of the present network as well as its capability for problems in irregular domains and in higher dimensions.

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“…They reformulated the interface problem as a least-squares problem and solved it by stochastic gradient de-scent. [22] proposed the discontinuity capturing shallow neural network (DCSNN) to approximate piecewise continuous functions and solved elliptic interface problems by minimizing the mean squared error loss consistent with the residual of the equation, boundary and interface jump conditions.…”

mentioning

confidence: 99%

Deep unfitted Nitsche method for elliptic interface problems

Guo,

Yang

2021

Preprint

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In this paper, we propose a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the efficiency and accuracy of the proposed method.

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A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems

Cited by 5 publications

References 20 publications

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

A Shallow Ritz Method for Elliptic Problems with Singular Sources

Deep unfitted Nitsche method for elliptic interface problems

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