scholarly journals A comparative investigation of neural networks in solving differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262199860
Author(s):  
Enze Shi ◽  
Chuanju Xu
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Comparative Investigation ◽  
Loss Functions ◽  
Approximation Properties ◽  
Function Fitting ◽  
Müntz Polynomials ◽  
The Neural Networks

Methods for solving differential equations based on neural networks have been widely proposed in recent years. However, limited open literature to date has reported the choice of loss functions and the hyperparameters of the network and how it influences the quality of numerical solutions. In the present work we intend to address this issue. Precisely we will focus on possible choices of loss functions and compare their efficiency in solving differential equations through a series of numerical experiments. In particular, a comparative investigation is performed between the natural neural networks and Ritz neural networks, with and without penalty for the boundary conditions. The sensitivity on the accuracy of the neural networks with respect to the size of training set, the number of nodes, and the penalty parameter is also studied. In order to better understand the training behavior of the proposed neural networks, we further investigate the approximation properties of the neural networks in function fitting. A particular attention is paid to approximating Müntz polynomials by neural networks.

scholarly journals A Numerical Method for Solving Fractional Differential Equations by Using Neural Network

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Haidong Qu ◽  
Xuan Liu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Initial Value ◽  
The Neural Networks ◽  
Fractional Differential

We present a new method for solving the fractional differential equations of initial value problems by using neural networks which are constructed from cosine basis functions with adjustable parameters. By training the neural networks repeatedly the numerical solutions for the fractional differential equations were obtained. Moreover, the technique is still applicable for the coupled differential equations of fractional order. The computer graphics and numerical solutions show that the proposed method is very effective.

scholarly journals ASSESSING THE SEMANTIC SIMILARITY OF IMAGES OF SILK FABRICS USING CONVOLUTIONAL NEURAL NETWORKS

2020 ◽  
Vol V-2-2020 ◽  
pp. 641-648
Author(s):  
D. Clermont ◽  
M. Dorozynski ◽  
D. Wittich ◽  
F. Rottensteiner
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Convolutional Neural Network ◽  
Image Features ◽  
Training Data ◽  
Loss Functions ◽  
Silk Fabrics ◽  
Semantic Properties ◽  
The Eu

Abstract. This paper proposes several methods for training a Convolutional Neural Network (CNN) for learning the similarity between images of silk fabrics based on multiple semantic properties of the fabrics. In the context of the EU H2020 project SILKNOW (http://silknow.eu/), two variants of training were developed, one based on a Siamese CNN and one based on a triplet architecture. We propose different definitions of similarity and different loss functions for both training strategies, some of them also allowing the use of incomplete information about the training data. We assess the quality of the trained model by using the learned image features in a k-NN classification. We achieve overall accuracies of 93–95% and average F1-scores of 87–92%.

Cosine Radial Basis Function Neural Networks for Solving Fractional Differential Equations

2017 ◽  
Vol 9 (3) ◽  
pp. 667-679 ◽  
Author(s):  
Haidong Qu
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Fractional Differential Equations ◽  
Basis Function ◽  
Numerical Solutions ◽  
Langevin Equations ◽  
Initial Value ◽  
Radial Basis ◽  
Fractional Differential

AbstractIn this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

scholarly journals Supporting the decision during inter-operational inspection of the electrodes based on the ensemble of neural networks

Mechanik ◽  
2018 ◽  
Vol 91 (12) ◽  
pp. 1060-1063
Author(s):  
Halina Nieciąg ◽  
Rafał Kudelski ◽  
Krzysztof Zagórski
Keyword(s):  
Neural Networks ◽  
Flexible Manufacturing ◽  
Electro Discharge Machining ◽  
Tool Monitoring ◽  
Geometrical Quality ◽  
The Neural Networks ◽  
Physical Phenomena ◽  
Random Nature ◽  
Edm Process

In this paper the method based on the ensemble of artificial neural networks is presented for prediction of the geometrical quality of workpieces after electro-discharge machining (EDM). The complexity and random nature of physical phenomena accompanying the EDM process excluded the theoretical ways. The working electrodes were measured using CMM in flexible manufacturing system. The data obtained from inter-operational measurements were used for the neural networks training. Commonly used measures to express the tool wear turn out to be useless due to their large uncertainty. The tool monitoring and the ensemble method provided more stable diagnosis of the condition of the tool.

scholarly journals Systems biology informed deep learning for inferring parameters and hidden dynamics

2019 ◽  
Author(s):  
Alireza Yazdani ◽  
Lu Lu ◽  
Maziar Raissi ◽  
George Em Karniadakis
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
Systems Biology ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
System Level ◽  
Model Parameters ◽  
Unknown Parameters ◽  
The Neural Networks ◽  
Noisy Measurements

AbstractMathematical models of biological reactions at the system-level lead to a set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a new systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, external forcing, and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems.Author summaryThe dynamics of systems biological processes are usually modeled using ordinary differential equations (ODEs), which introduce various unknown parameters that need to be estimated efficiently from noisy measurements of concentration for a few species only. In this work, we present a new “systems-informed neural network” to infer the dynamics of experimentally unobserved species as well as the unknown parameters in the system of equations. By incorporating the system of ODEs into the neural networks, we effectively add constraints to the optimization algorithm, which makes the method robust to noisy and sparse measurements.

Novel U-net based deep neural networks for transmission tomography

2021 ◽  
pp. 1-19
Author(s):  
Csaba Olasz ◽  
László G. Varga ◽  
Antal Nagy
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Tomographic Reconstruction ◽  
Similarity Index ◽  
Simulated Data ◽  
Electrical Noise ◽  
Projection Data ◽  
Beam Hardening ◽  
The Neural Networks

BACKGROUND: The fusion of computer tomography and deep learning is an effective way of achieving improved image quality and artifact reduction in reconstructed images. OBJECTIVE: In this paper, we present two novel neural network architectures for tomographic reconstruction with reduced effects of beam hardening and electrical noise. METHODS: In the case of the proposed novel architectures, the image reconstruction step is located inside the neural networks, which allows the network to be trained by taking the mathematical model of the projections into account. This strong connection enables us to enhance the projection data and the reconstructed image together. We tested the two proposed models against three other methods on two datasets. The datasets contain physically correct simulated data, and they show strong signs of beam hardening and electrical noise. We also performed a numerical evaluation of the neural networks on the reconstructed images according to three error measurements and provided a scoring system of the methods derived from the three measures. RESULTS: The results showed the superiority of the novel architecture called TomoNet2. TomoNet2 improved the quality of the images according to the average Structural Similarity Index from 0.9372 to 0.9977 and 0.9519 to 0.9886 on the two data sets, when compared to the FBP method. This network also yielded the best results for 79.2 and 53.0 percent for the two datasets according to Peak-Signal-to-Noise-Ratio compared to the other improvement techniques. CONCLUSIONS: Our experimental results showed that the reconstruction step used in skip connections in deep neural networks improves the quality of the reconstructions. We are confident that our proposed method can be effectively applied to other datasets for tomographic purposes.

scholarly journals Application of Neural Networks in Distribution of the Load in Cluster-Based Web Systems

2021 ◽  
Vol 12 (1) ◽  
pp. 79
Author(s):  
Waldemar Pokuta ◽  
Krzysztof Zatwarnicki
Keyword(s):  
Neural Networks ◽  
Quality Of Service ◽  
Load Distribution ◽  
Production Systems ◽  
Computing Systems ◽  
Resource Configuration ◽  
Artificial Neuron ◽  
The Neural Networks ◽  
Web Systems

Cloud computing systems revolutionized the Internet, and web systems in particular. Quality of service is the basis of resource configuration management in the cloud. Load balancing mechanisms are implemented in order to reduce costs and increase the quality of service. The usage of those methods with adaptive intelligent algorithms can deliver the highest quality of service. In this article, the method of load distribution using neural networks to estimate service times is presented. The discussed and conducted research and experiments include many approaches, among others, application of a single artificial neuron, different structures of the neural networks, and different inputs for the networks. The results of the experiments let us choose a solution that enables effective load distribution in the cloud. The best solution is also compared with other intelligent approaches and distribution methods often used in production systems.

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