A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks

Xie, Haiyang; Zhai, Chao; Liu, Li; Huang, Yong

doi:10.48550/arxiv.2205.06658

Cited by 2 publications

(3 citation statements)

References 37 publications

(41 reference statements)

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“…Which leads overall to a worse approximation of the real solution. To avoid this problem, in [164] a special weight function is introduced. The function is constructed in such a way, that the weight is zero at the interfaces where the discontinuities are and O(1) away from the interfaces.…”

Section: Setting Of Numerical Implementationsmentioning

confidence: 99%

Deep learning methods for partial differential equations and related parameter identification problems

Nganyu Tanyu,

Ning,

Freudenberg

et al. 2023

Inverse Problems

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Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

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Section: Setting Of Numerical Implementationsmentioning

confidence: 99%

Deep learning methods for partial differential equations and related parameter identification problems

Nganyu Tanyu,

Ning,

Freudenberg

et al. 2023

Inverse Problems

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“…One can train the points in general regions first (which are often easy to train) by embed weight functions into the loss function. Similar, a strategy that training the general regions (simple to train) first and then gradually forcing the networks predicting well in the key regions (hard to train) can be considered [32][33]. In [32], a weighted equations (WE) method with PINNs is used improve the shock capturing ability.…”

Section: Introductionmentioning

confidence: 99%

“…In [32], a weighted equations (WE) method with PINNs is used improve the shock capturing ability. In [33], a weighted first-order formulation was established to solve anisotropic diffusion equations.…”

Section: Introductionmentioning

confidence: 99%

Gradient-weighted physics-informed neural networks for one-dimensional Euler equation

Xiong¹,

Liu²,

Liu³

et al. 2022

Preprint

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<p> Physics informed neural networks (PINNs) have been a well-known tool in solving forward and inverse partial differential equations (PDEs) problem. In particular, for the forward problem, only the initial/boundary conditions and the equation itself as the physical information are needed to obtain the predicted solution within the definition domain. However, some existing works show that the standard PINN method is insufficient for solving complex problems. For the relatively complex one-dimensional Euler equation problem, due to the limited accuracy of the standard PINN method, the algorithm is improved in two aspects in this study. First, a gradient-related weight function is introduced into the loss function and the framework of gradient-weighted physics-informed neural networks (gwPINNs) is proposed. In particular, the weight function is carefully designed. Second, the sequence-to-sequence (seq2seq) learning is applied in the training process to improve the precision. Then, four cases of Riemann problems are solved using the improved algorithm. The results show that the improved algorithm can obtain high accuracy in solving these problems, and verify the feasibility of solving one-dimensional Euler equation based on the improved PINN method.</p>

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A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks

Cited by 2 publications

References 37 publications

Deep learning methods for partial differential equations and related parameter identification problems

Deep learning methods for partial differential equations and related parameter identification problems

Gradient-weighted physics-informed neural networks for one-dimensional Euler equation

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