scholarly journals Deep neural network modeling of unknown partial differential equations in nodal space

2021 ◽  
pp. 110782
Author(s):  
Zhen Chen ◽  
Victor Churchill ◽  
Kailiang Wu ◽  
Dongbin Xiu
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Network ◽  
Network Modeling ◽  
Neural Network Modeling ◽  
Partial Differential

scholarly journals A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 136
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke ◽  
Michaela Szölgyenyi
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Network ◽  
Risk Measure ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Control Problems ◽  
Partial Differential ◽  
Neural Network Algorithm

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

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>

A study on global optimization and deep neural network modeling method in performance-seeking control

2019 ◽  
Vol 234 (1) ◽  
pp. 46-59 ◽  
Author(s):  
Qiangang Zheng ◽  
Dawei Fu ◽  
Yong Wang ◽  
Haoying Chen ◽  
Haibo Zhang
Keyword(s):  
Neural Network ◽  
Genetic Algorithm ◽  
Global Optimization ◽  
Interval Analysis ◽  
Sequential Quadratic Programming ◽  
Deep Neural Network ◽  
Network Modeling ◽  
Engine Performance ◽  
Neural Network Modeling ◽  
Swarm Optimization

In this article, a novel performance-seeking control method based on deep neural network and interval analysis is proposed to obtain a better engine performance. A deep neural network modeling method which has stronger representation capability than conventional neural network and can deal with big training data is adopted to establish an on-board model in the subsonic and supersonic cruising envelops. Meanwhile, a global optimization algorithm interval analysis is applied here to get a better engine performance. Finally, two simulation experiments are conducted to verify the effectiveness of the proposed methods. One is the on-board model modeling which compares the deep neural network with the conventional neural network, and the other is the performance-seeking control simulations comparing interval analysis with feasible sequential quadratic programming, particle swarm optimization, and genetic algorithm, respectively. These two experiments show that the deep neural network has much higher precision than the conventional neural network and the interval analysis gets much better engine performance than feasible sequential quadratic programming, particle swarm optimization, and genetic algorithm.

Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

2020 ◽  
Vol 29 (05) ◽  
pp. 2050009
Author(s):  
Pola Lydia Lagari ◽  
Lefteri H. Tsoukalas ◽  
Salar Safarkhani ◽  
Isaac E. Lagaris
Keyword(s):  
Neural Network ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Systematic Approach ◽  
Thermal Contact ◽  
Initial Boundary ◽  
Spatial Domain ◽  
Interface Conditions ◽  
Partial Differential

A systematic approach is developed for constructing proper trial solutions to Partial Differential Equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. The spatial domain considered is of the rectangular hyper-box type. On each face either Dirichlet or Neumann conditions may apply. Robin conditions may be accommodated as well. Interface conditions that induce discontinuities, have not been treated to date in the relevant neural network literature. As an illustration a common problem of heat conduction through a system of two rods in thermal contact is considered.

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