scholarly journals Solving parametric PDE problems with artificial neural networks

2020 ◽  
pp. 1-15
Author(s):  
YUEHAW KHOO ◽  
JIANFENG LU ◽  
LEXING YING
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Evolution ◽  
Random Coefficients ◽  
Numerical Partial Differential Equations ◽  
The Neural Network ◽  
Physical Quantities

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of the coefficient fields. Based on such observation, we propose using neural network to parameterise the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural network can be justified by viewing the neural network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

scholarly journals Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Idris Kharroubi ◽  
Thomas Lim ◽  
Xavier Warin
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Learning Approach ◽  
The Neural Network ◽  
Machine Learning Approach ◽  
Mesh Grid

AbstractWe study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments.

scholarly journals Artificial neural networks applied for soil class prediction in mountainous landscape of the Serra do Mar¹

2014 ◽  
Vol 38 (6) ◽  
pp. 1681-1693 ◽  
Author(s):  
Braz Calderano Filho ◽  
Helena Polivanov ◽  
César da Silva Chagas ◽  
Waldir de Carvalho Júnior ◽  
Emílio Velloso Barroso ◽  
...  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Kappa Index ◽  
Class Prediction ◽  
Soil Class ◽  
Serra Do Mar ◽  
The Neural Network ◽  
Artificial Neural ◽  
Local Geology

Soil information is needed for managing the agricultural environment. The aim of this study was to apply artificial neural networks (ANNs) for the prediction of soil classes using orbital remote sensing products, terrain attributes derived from a digital elevation model and local geology information as data sources. This approach to digital soil mapping was evaluated in an area with a high degree of lithologic diversity in the Serra do Mar. The neural network simulator used in this study was JavaNNS and the backpropagation learning algorithm. For soil class prediction, different combinations of the selected discriminant variables were tested: elevation, declivity, aspect, curvature, curvature plan, curvature profile, topographic index, solar radiation, LS topographic factor, local geology information, and clay mineral indices, iron oxides and the normalized difference vegetation index (NDVI) derived from an image of a Landsat-7 Enhanced Thematic Mapper Plus (ETM+) sensor. With the tested sets, best results were obtained when all discriminant variables were associated with geological information (overall accuracy 93.2 - 95.6 %, Kappa index 0.924 - 0.951, for set 13). Excluding the variable profile curvature (set 12), overall accuracy ranged from 93.9 to 95.4 % and the Kappa index from 0.932 to 0.948. The maps based on the neural network classifier were consistent and similar to conventional soil maps drawn for the study area, although with more spatial details. The results show the potential of ANNs for soil class prediction in mountainous areas with lithological diversity.

Improving Pin-Fin Heat Transfer Predictions Using Artificial Neural Networks

2013 ◽  
Author(s):  
Jason K. Ostanek
Keyword(s):  
Neural Network ◽  
Heat Transfer ◽  
Neural Networks ◽  
Reynolds Number ◽  
Artificial Neural Networks ◽  
Power Law ◽  
Transfer Data ◽  
Pin Fin ◽  
The Neural Network ◽  
Artificial Neural

In much of the public literature on pin-fin heat transfer, Nusselt number is presented as a function of Reynolds number using a power-law correlation. Power-law correlations typically have an accuracy of 20% while the experimental uncertainty of such measurements is typically between 5% and 10%. Additionally, the use of power-law correlations may require many sets of empirical constants to fully characterize heat transfer for different geometrical arrangements. In the present work, artificial neural networks were used to predict heat transfer as a function of streamwise spacing, spanwise spacing, pin-fin height, Reynolds number, and row position. When predicting experimental heat transfer data, the neural network was able to predict 73% of array-averaged heat transfer data to within 10% accuracy while published power-law correlations predicted 48% of the data to within 10% accuracy. Similarly, the neural network predicted 81% of row-averaged data to within 10% accuracy while 52% of the data was predicted to within 10% accuracy using power-law correlations. The present work shows that first-order heat transfer predictions may be simplified by using a single neural network model rather than combining or interpolating between power-law correlations. Furthermore, the neural network may be expanded to include additional pin-fin features of interest such as fillets, duct rotation, pin shape, pin inclination angle, and more making neural networks expandable and adaptable models for predicting pin-fin heat transfer.

Artificial neural networks in the problem of determining the position and size of the solar disk in wide-angle photographs of the sky

2021 ◽  
Author(s):  
Mikhail Borisov ◽  
Mikhail Krinitskiy
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Solar Disk ◽  
Weather Conditions ◽  
Wide Angle ◽  
Know How ◽  
The Neural Network ◽  
Artificial Neural ◽  
The Moment

<p>Total cloud score is a characteristic of weather conditions. At the moment, there are algorithms that automatically calculate cloudiness based on a photograph of the sky These algorithms do not know how to find the solar disk, so their work is not absolutely accurate.</p><p>To create an algorithm that solves this data, the data used, obtained as a result of sea research voyages, is used, which is marked up for training the neural network.</p><p>As a result of the work, an algorithm was obtained based on neural networks, based on a photograph of the sky, in order to determine the size and position of the solar disk, other algorithms can be used to work with images of the visible hemisphere of the sky.</p>

Applying neural network for identification of land surface model parameters

2021 ◽  
Author(s):  
Ruslan Chernyshev ◽  
Mikhail Krinitskiy ◽  
Viktor Stepanenko
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermodynamic Model ◽  
Land Surface ◽  
Land Surface Model ◽  
Surface Model ◽  
Partial Differential

<p>This work is devoted to development of neural networks for identification of partial differential equations (PDE) solved in the land surface scheme of INM RAS Earth System model (ESM). Atmospheric and climate models are in the top of the most demanding for supercomputing resources among research applications. Spatial resolution and a multitude of physical parameterizations used in ESMs continuously increase. Most of parameters are still poorly constrained, many of them cannot be measured directly. To optimize model calibration time, using neural networks looks a promising approach. Neural networks are already in wide use in satellite imaginary (Su Jeong Lee, et al, 2015; Krinitskiy M. et al, 2018) and for calibrating parameters of land surface models (Yohei Sawada el al, 2019). Neural networks have demonstrated high efficiency in solving conventional problems of mathematical physics (Lucie P. Aarts el al, 2001; Raissi M. et al, 2020). </p><p>We develop a neural networks for optimizing parameters of nonlinear soil heat and moisture transport equation set. For developing we used Python3 based programming tools implemented on GPUs and Ascend platform, provided by Huawei. Because of using hybrid approach combining neural network and classical thermodynamic equations, the major purpose was finding the way to correctly calculate backpropagation gradient of error function, because model trains and is being validated on the same temperature data, while model output is heat equation parameter, which is typically not known. Neural network model has been runtime trained using reference thermodynamic model calculation with prescribed parameters, every next thermodynamic model step has been used for fitting the neural network until it reaches the loss function tolerance.</p><p>Literature:</p><p>1.     Aarts, L.P., van der Veer, P. “Neural Network Method for Solving Partial Differential Equations”. Neural Processing Letters 14, 261–271 (2001). https://doi.org/10.1023/A:1012784129883</p><p>2.     Raissi, M., P. Perdikaris and G. Karniadakis. “Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations.” ArXiv abs/1711.10561 (2017): n. pag.</p><p>3.     Lee, S.J., Ahn, MH. & Lee, Y. Application of an artificial neural network for a direct estimation of atmospheric instability from a next-generation imager. Adv. Atmos. Sci. 33, 221–232 (2016). https://doi.org/10.1007/s00376-015-5084-9</p><p>4.     Krinitskiy M, Verezemskaya P, Grashchenkov K, Tilinina N, Gulev S, Lazzara M. Deep Convolutional Neural Networks Capabilities for Binary Classification of Polar Mesocyclones in Satellite Mosaics. Atmosphere. 2018; 9(11):426.</p><p>5.     Sawada, Y.. “Machine learning accelerates parameter optimization and uncertainty assessment of a land surface model.” ArXiv abs/1909.04196 (2019): n. pag.</p><p>6.     Shufen Pan et al. Evaluation of global terrestrial evapotranspiration using state-of-the-art approaches in remote sensing, machine learning and land surface modeling. Hydrol. Earth Syst. Sci., 24, 1485–1509 (2020)</p><p>7.     Chaney, Nathaniel & Herman, Jonathan & Ek, M. & Wood, Eric. (2016). Deriving Global Parameter Estimates for the Noah Land Surface Model using FLUXNET and Machine Learning: Improving Noah LSM Parameters. Journal of Geophysical Research: Atmospheres. 121. 10.1002/2016JD024821.</p><p> </p><p> </p>

Weights Direct Determination of Feedforward Neural Networks without Iterative BP-Training

2010 ◽  
pp. 197-225 ◽  
Author(s):  
Yunong Zhang ◽  
Ning Tan
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Direct Determination ◽  
Bp Neural Networks ◽  
Learning Rates ◽  
The Neural Network ◽  
Gradient Based ◽  
Artificial Neural ◽  
Output Behavior

Artificial neural networks (ANN), especially with error back-propagation (BP) training algorithms, have been widely investigated and applied in various science and engineering fields. However, the BP algorithms are essentially gradient-based iterative methods, which adjust the neural-network weights to bring the network input/output behavior into a desired mapping by taking a gradient-based descent direction. This kind of iterative neural-network (NN) methods has shown some inherent weaknesses, such as, 1) the possibility of being trapped into local minima, 2) the difficulty in choosing appropriate learning rates, and 3) the inability to design the optimal or smallest NN-structure. To resolve such weaknesses of BP neural networks, we have asked ourselves a special question: Could neural-network weights be determined directly without iterative BP-training? The answer appears to be YES, which is demonstrated in this chapter with three positive but different examples. In other words, a new type of artificial neural networks with linearly-independent or orthogonal activation functions, is being presented, analyzed, simulated and verified by us, of which the neural-network weights and structure could be decided directly and more deterministically as well (in comparison with usual conventional BP neural networks).

Artificial Neural Networks

2011 ◽  
pp. 222-243
Author(s):  
Joarder Kamruzzaman ◽  
Ruhul Sarker
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Network ◽  
Artificial Neural Networks ◽  
Network Models ◽  
Neural Network Models ◽  
Network Applications ◽  
The Neural Network ◽  
Artificial Neural ◽  
Neural Network Applications

The primary aim of this chapter is to present an overview of the artificial neural network basics and operation, architectures, and the major algorithms used for training the neural network models. As can be seen in subsequent chapters, neural networks have made many useful contributions to solve theoretical and practical problems in finance and manufacturing areas. The secondary aim here is therefore to provide a brief review of artificial neural network applications in finance and manufacturing areas.

Artificial Neural Networks and Learning Techniques

2016 ◽  
pp. 89-112
Author(s):  
Pushpendu Kar ◽  
Anusua Das
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Research Work ◽  
The Neural Network ◽  
Learning Techniques ◽  
Learning Procedure ◽  
Processing Component ◽  
Artificial Neural ◽  
Artificial Neural Network Ann

The recent craze for artificial neural networks has spread its roots towards the development of neuroscience, pattern recognition, machine learning and artificial intelligence. The theoretical neuroscience is basically converging towards the basic concept that the brain acts as a complex and decentralized computer which can perform rigorous calculations in a different approach compared to the conventional digital computers. The motivation behind the study of neural networks is due to their similarity in the structure of the human central nervous system. The elementary processing component of an Artificial Neural Network (ANN) is called as ‘Neuron'. A large number of neurons interconnected with each other mimic the biological neural network and form an ANN. Learning is an inevitable process that can be used to train an ANN. We can only transfer knowledge to the neural network by the learning procedure. This chapter presents the detailed concepts of artificial neural networks in addition to some significant aspects on the present research work.

Contributions to Predict the Malfunction Probability of the Human-Machine-Environment System, Using Artificial Neural Networks

2015 ◽  
Vol 760 ◽  
pp. 771-776
Author(s):  
Daniel Constantin Anghel ◽  
Nadia Belu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Design Experiment ◽  
Manufacturing Processes ◽  
Feed Forward Neural Network ◽  
Feed Forward ◽  
Linear Relationships ◽  
The Neural Network ◽  
Artificial Neural

This paper presents the application of Artificial Neural Networks to predict the malfunction probability of the human-machine-environment system, in order to provide some guidance to designers of manufacturing processes. Artificial Neural Networks excel in gathering difficult non-linear relationships between the inputs and outputs of a system. We used, in this work, a feed forward neural network in order to predict the malfunction probability. The neural network is simulated with Matlab. The design experiment presented in this paper was realized at University of Pitesti, at the Faculty of Mechanics and Technology, Technology and Management Department.

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