An Artificial Neural Networks Method for Solving Partial Differential Equations

Alharbi, Abir

doi:10.1063/1.3498013

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…This approach is called the Hopfield Finite Difference method (HFD), and it has the advantage of working in a parallel mode and giving faster accurate results. The HFD method presented solutions to the classical Wave, Heat (Diffusion), Poisson equations [8,9], and to systems of PDEs [10].…”

Section: Hopfield-finite-difference Methodsmentioning

confidence: 99%

A solution to neural field equations by a recurrent neural network method

Alharbi¹

2012

AIP Conference Proceedings

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Neural field equations (NFE) are used to model the activity of neurons in the brain, it is introduced from a single neuron 'integrate-and-fire model' starting point. The neural continuum is spatially discretized for numerical studies, and the governing equations are modeled as a system of ordinary differential equations. In this article the recurrent neural network approach is used to solve this system of ODEs. This consists of a technique developed by combining the standard numerical method of finite-differences with the Hopfield neural network. The architecture of the net, energy function, updating equations, and algorithms are developed for the NFE model. A Hopfield Neural Network is then designed to minimize the energy function modeling the NFE. Results obtained from the Hopfield-finite-differences net show excellent performance in terms of accuracy and speed. The parallelism nature of the Hopfield approaches may make them easier to implement on fast parallel computers and give them the speed advantage over the traditional methods.

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Section: Hopfield-finite-difference Methodsmentioning

confidence: 99%

A solution to neural field equations by a recurrent neural network method

Alharbi¹

2012

AIP Conference Proceedings

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“…This approach is called the Hopfield Finite Difference method (HFD), and it has the advantage of working in a parallel mode and giving fast and accurate results. The HFD method has been used to solve the classical Wave, Heat (Diffusion), Poisson equations (Alharbi, 1997(Alharbi, , 2010(Alharbi, , 2012, and to systems of PDEs (Alharbi and Alahmadi, 2008). We will use the HFD to solve the point source diffusion equation in the ECS described in the last section.…”

Section: The Diffusion Equation In Ecs By the Hopfield Neural Networkmentioning

confidence: 99%

Using a Hopfield Iterative Neural Network to Explain Diffusion in the Brain’s Extracellular Space Structure

Alharbi

2014

Proceedings of the International Conference on Neural Computation Theory and Applications

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Many therapies for drug delivery to the brain are based on diffusion, and diffusion in this extracellular space is based on micro-techniques that can be modelled with classical differential equations such as the point source diffusion equation. In this paper an energy function is constructed using a finite-difference approximation to the governing diffusion equation and then minimized by a Hopfield neural network. The synergy of Hopfield neural networks with finite difference approximation is promising. The neural network approach is capable of giving insight to the complex brain activity better than any other classical numerical method and the parallelism nature of the Hopfield neural networks approach is easier to implement on fast parallel computers and this will make them faster than the traditional methods for modelling this complex problem. Moreover, the effect of the involved parameters on the diffusion distribution and drug delivery in the ECS is investigated.

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“…Coupled with the automatic differentiation technique, some studies have shown that neural networks can be applied to solve the PDEs [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Besides, many effective algorithms are proposed to solve some high-dimensional PDEs in [58][59][60][61][62][63][64][65]. Sirignano and Spiliopoulos [36] proposed the DGM to address the curse of dimensionality problem when solving high-dimensional PDEs.…”

Section: Introductionmentioning

confidence: 99%

A Deep Learning Galerkin Method for the Closed-Loop Geothermal System

Zhang¹,

Li²

2022

Preprint

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There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled multi-physics PDEs and mainly consists of a framework of underground heat exchange pipelines to extract the geothermal heat from the geothermal reservoir. This method is a natural combination of Galerkin Method and machine learning with the solution approximated by a neural network instead of a linear combination of basis functions. We train the neural network by randomly sampling the spatiotemporal points and minimize loss function to satisfy the differential operators, initial condition, boundary and interface conditions. Moreover, the approximate ability of the neural network is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in L 2 norm under certain conditions. Finally, some numerical examples are carried out to demonstrate the approximation ability of the neural networks intuitively.

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An Artificial Neural Networks Method for Solving Partial Differential Equations

Cited by 4 publications

References 2 publications

A solution to neural field equations by a recurrent neural network method

A solution to neural field equations by a recurrent neural network method

Using a Hopfield Iterative Neural Network to Explain Diffusion in the Brain’s Extracellular Space Structure

A Deep Learning Galerkin Method for the Closed-Loop Geothermal System

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