Fractional Chebyshev deep neural network (FCDNN) for solving differential models

Hajimohammadi, Zeinab; Baharifard, Fatemeh; Ghodsi, Ali; Parand, Kourosh

doi:10.1016/j.chaos.2021.111530

Cited by 16 publications

(3 citation statements)

References 69 publications

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“…GCN is a semi‐supervised model that can deal with graph structures which is an advancement of CNN in the field of graph structures. In recent years, GCN have received extensive attention and GCN layers have also undergone significant improvements, such as spectral graph [38], first‐order filters [39] and Chebyshev polynomial [40]. GCN has been widely utilized in various fields due to its powerful ability to extract spatial information.…”

Section: Methodsmentioning

confidence: 99%

A deep neural network model with GCN and 3D convolutional network for short‐term metro passenger flow forecasting

Zhang

Wang

Chen

et al. 2023

IET Intelligent Trans Sys

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Rail transit has many advantages, such as large passenger capacity, convenience, safety, and environmental protection, making it the preferred travel mode for most passengers. Deep learning has become an effective method for short‐term rail passenger flow prediction. A deep learning model which combines a graph convolutional network and a three‐dimensional Convolutional Neural Network improved by a residual module and an attention mechanism (ARConv‐CGN) is proposed. First, historical passenger inflow and outflow data are aggregated into three patterns: recent pattern, daily pattern, and weekly pattern separately. The GCN is applied to capture the spatiotemporal and topological information of passenger flows in each pattern. Second, a three‐dimensional convolutional neural network is used to deeply integrate three patterns of passenger flow information. Additionally, a residual module is used to increase the number of neural network layers and prevent the gradient of the deep neural network from disappearing. Finally, an attention mechanism is also introduced to adjust the importance of passenger flow in different pattern, so that improving the performance of the model. Training with passenger flow data from automatic fare collection on weekdays in Beijing and Xiamen provides a good demonstration of the superiority of ARConv‐CGN in metro passenger flow forecasting.

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Section: Methodsmentioning

confidence: 99%

A deep neural network model with GCN and 3D convolutional network for short‐term metro passenger flow forecasting

Zhang

Wang

Chen

et al. 2023

IET Intelligent Trans Sys

View full text Add to dashboard Cite

show abstract

“…In [31], the authors employed a deep learning library for solving differential equations. Besides, the authors in [32], provided a deep neural network with Fractional Chebyshev polynomials as operational nodes and used fractional calculations and Chebyshev Gaussian integration to solve Fractional Fredholm integral equations.…”

Section: Deep Neural Networkmentioning

confidence: 99%

A New Numerical Method for Solving Neuro-Cognitive Models via Chebyshev Deep Neural Network (CDNN)

Mohammadi

Babaei

Hajimohammadi

et al. 2023

Preprint

Self Cite

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One of the fundamental applications of artificial neural networks is solving Partial Differential Equations (PDEs) which has been considered in this paper. We have created an effective method by combining the spectral methods and multi-layer perceptron to solve Generalized Fitzhugh–Nagumo (GFHN) equation. In this method, we have used Chebyshev polynomials as activation functions of the multi-layer perceptron. In order to solve PDEs, independent variables, which are collocation points, have been used as input dataset. Furthermore, the loss function has been constructed from the residual of the equation and its boundary condition. Minimizing the loss function has adjusted the appropriate values for the parameters of the network. Hence, the network has shown an outstanding performance not only on the training dataset but also on the Unseen data. Some numerical examples and a comparison between the results of our proposed method and other existing approaches have been provided to show the efficiency and accuracy of the proposed method. For this purpose different cases such as linear, nonlinear and multi dimensional equations are considered.

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“…So the solutions of the equations become more singular, which makes challenging to learn such PDEs. There are only few deep-learning-based models aimed at fractional equations [10,28], and the computation of fractional Laplacian for the neural network function is based on the finite difference method, which is highly timeconsuming. The third challenge relates to the inefficiency on hyperparameter tuning of PINNs [20,31,40].…”

Section: Introductionmentioning

confidence: 99%

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

Zhang¹,

Hu²,

Meng³

et al. 2022

Preprint

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Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. Due to the importance of Navier-Stokes equations, the development on efficient numerical schemes is important for both science and engineer. Recently, with the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations, which can accelerate the simulation or inferring process in a mesh-free and differentiable way. In this paper, we point out that the capability of existing deep Navier-Stokes informed methods is limited to handle non-smooth or fractional equations, which are two critical situations in reality. To this end, we propose the Deep Random Vortex Method (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation. Specifically, the random vortex dynamics motivates a Monte Carlo based loss function for training the neural network, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVM not only can efficiently solve Navier-Stokes equations involving rough path, nondifferentiable initial conditions and fractional operators, but also inherits the mesh-free and differentiable benefits of the deep-learning-based solver. We conduct experiments on the Cauchy problem, parametric solver learning, and the inverse problem of both 2-d and 3-d incompressible Navier-Stokes equations. The proposed method achieves accurate results for simulation and inference of Navier-Stokes equations. Especially for the cases that include singular initial conditions, DRVM significantly outperforms existing PINN method.

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Fractional Chebyshev deep neural network (FCDNN) for solving differential models

Cited by 16 publications

References 69 publications

A deep neural network model with GCN and 3D convolutional network for short‐term metro passenger flow forecasting

A deep neural network model with GCN and 3D convolutional network for short‐term metro passenger flow forecasting

A New Numerical Method for Solving Neuro-Cognitive Models via Chebyshev Deep Neural Network (CDNN)

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

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