Solving stiff ordinary differential equations and partial differential equations using analog computing based on cellular neural networks

2009 ◽  
Author(s):  
J.C. Chedjou ◽  
K. Kyamakya ◽  
M.A. Latif ◽  
U.A. Khan ◽  
I. Moussa ◽  
...  
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Partial Differential ◽  
Stiff Ordinary Differential Equations ◽  
Analog Computing

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

CHAOS AND HYPERCHAOS IN A CLASS OF SIMPLE CELLULAR NEURAL NETWORKS MODELED BY O.D.E.

2006 ◽  
Vol 16 (09) ◽  
pp. 2729-2736 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Exponents ◽  
Cellular Neural Networks ◽  
New Class ◽  
Connection Matrices ◽  
Low Dimensional

This paper presents a new class of chaotic and hyperchaotic low dimensional cellular neural networks modeled by ordinary differential equations with some simple connection matrices. The chaoticity of these neural networks is indicated by positive Lyapunov exponents calculated by a computer.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

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>

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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