Numerical solution for high-dimensional partial differential equations based on deep learning with residual learning and data-driven learning

Zhang

Journal of Healthcare Engineering

et al. 2021

In order to study the influence of quantitative magnetic susceptibility mapping (QSM) on them. A 2.5D Attention U-Net Network based on multiple input and multiple output, a method for segmenting RN, SN, and STN regions in high-resolution QSM images is proposed, and deep learning realizes accurate segmentation of deep nuclei in brain QSM images. Experimental results show data first cuts each layer of 0 100 case data, based on the image center, from 384 × 288 to the size of 128 × 128. Image combination: each layer of the image in the layer direction combines with two adjacent images into a 2.5D image, i.e., (It − m It; It + i), where It represents the layer i image. At this time, the size of the image changes from 128 × 128 to 128 × 128 × 3, in which 3 represents three consecutive layers of images. The SNR of SWP I to STN is twice that of SWI. The small deep gray matter nuclei (RN, SN, and STN) in QSM images of the brain and the pancreas with irregular shape and large individual differences in abdominal CT images can be automatically segmented.

Section: Introductionmentioning

confidence: 99%

The Effect of Deep Learning-Based QSM Magnetic Resonance Imaging on the Subthalamic Nucleus

Zhang

Journal of Healthcare Engineering

et al. 2021

“…Partial differential equations (PDEs) have a wide range of applications, from the simulation of seismic waves on Earth to the spread of infectious diseases through human populations. Engineers, scientists, and mathematicians have resorted to PDEs to describe complex phenomena involving many independent variables [1,2]. However, the process of solving PDEs is extremely difficult, traditional numerical methods are very complex and require a lot of computation.…”

Section: Introductionmentioning

confidence: 99%

Interaction phenomenon for variable coefficient Kadomtsev-Petviashvili equation by utilizing variable coefficient bilinear neural network method

Liu¹,

Zhu²

2023

Preprint

In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with variable coefficients is investigated by using the variable coefficient Bilinear neural network method. By establishing "3-2-2-1" and "3-3-2-1" models respectively, rich analytical solutions of the variable coefficient Kadomstev-Petviashvili equation are obtained. By choosing some special values of the parameters, the dynamics properties are demonstrated in some three-dimensional and density graphics.

“…, 2021; Famelis and Kaloutsa, 2021; Günel and Gör, 2021; Li and Wang, 2021; Schiassi et al. , 2021) and deep learning (Wang et al. , 2021; Weinan et al.…”

Section: Introductionmentioning

confidence: 99%

“…With the rapid development of computer hardware and various new theories, neural networks and other machine learning algorithms are gradually applied to various areas (Weng et al, 2021(Weng et al, , 2022. In fact, various machine learning algorithms such as least squares support vector machines (Lu et al, 2019;Mehrkanoon et al, 2012;Mehrkanoon and Suykens, 2015), neural networks (Eskiizmirliler et al, 2021;Famelis and Kaloutsa, 2021;G€ unel and G€ or, 2021;Li and Wang, 2021;Schiassi et al, 2021) and deep learning (Wang et al, 2021;Weinan et al, 2017) have been utilized to solve these differential equations. In particular, artificial neural networks (ANNs) are often superior to the traditional numerical methods on the field of solving differential equation Chakraverty, 2014, 2016;Yazdi et al, 2011Yazdi et al, , 2012Yazdi and Pourreza, 2010).…”

mentioning

confidence: 99%

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Weng

Sun

2022

Self Cite

PurposeThis paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).Design/methodology/approachThe proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.FindingsTo demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.Originality/valueThe H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.