Artificial neural network approach for solving fuzzy differential equations

The current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed-forward neural networks. First, the fuzzy differential equation is replaced by a system of ordinary differential equations. A trial solution of this system is written as a sum of two parts. The first part satisfies the initial condition and contains no adjustable parameters. The second part involves a feed-forward neural network containing adjustable parameters (the weights). Hence by construction, the initial condition is satisfied and the network is trained to satisfy the differential equations. This method, in comparison with existing numerical methods, shows that the use of neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.

[1]  Congxin Wu,et al.  Existence Theorem to the Cauchy Problem of Fuzzy Differential Equations under Compactness-Type Conditions , 1998, Inf. Sci..

[2]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[3]  Esmail Babolian,et al.  Numerically solution of fuzzy differential equations by Adomian method , 2004, Appl. Math. Comput..

[4]  Tofigh Allahviranloo,et al.  Numerical solution of fuzzy differential equations by predictor-corrector method , 2007, Inf. Sci..

[5]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[6]  Shiji Song,et al.  Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations , 2000, Fuzzy Sets Syst..

[7]  R. M. Jafelice,et al.  Fuzzy modeling in symptomatic HIV virus infected population , 2004, Bulletin of mathematical biology.

[8]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[9]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU) - an outline , 2005, GrC.

[10]  B. C. Brookes,et al.  Information Sciences , 2020, Cognitive Skills You Need for the 21st Century.

[11]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[12]  Luciano Stefanini,et al.  Parametric representation of fuzzy numbers and application to fuzzy calculus , 2006, Fuzzy Sets Syst..

[13]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential and integral equations , 1999, Fuzzy Sets Syst..

[14]  Kevin D. Reilly,et al.  Simulating continuous fuzzy systems , 2005, Inf. Sci..

[15]  M. Puri,et al.  DIFFERENTIAL FOR FUZZY FUNCTION , 1983 .

[16]  Y Chalco Cano,et al.  ON NEW SOLUTIONS OF FUZZY DIFFERENTIAL EQUATIONS , 2008 .

[17]  Saeid Abbasbandy,et al.  Numerical methods forfuzzy differential inclusions , 2004 .

[18]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[19]  Yurilev Chalco-Cano,et al.  Fuzzy differential equations and the extension principle , 2007, Inf. Sci..

[20]  James J. Buckley,et al.  Fuzzy differential equations , 2000, Fuzzy Sets Syst..

[21]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[22]  T Alah Viranlou,et al.  NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL EQUATION BY RUNGE-KUTTA METHOD , 2001 .

[23]  Saeid Abbasbandy,et al.  NUMERICAL METHODS FOR FUZZY DIFFERENTIAL INCLUSIONS , 2004 .

[24]  R. Bassanezi,et al.  Fuzzy modelling in population dynamics , 2000 .

[25]  Lotfi A. Zadeh,et al.  On Fuzzy Mapping and Control , 1996, IEEE Trans. Syst. Man Cybern..

[26]  Eyke Hüllermeier.  Numerical Methods for Fuzzy Initial Value Problems , 1999, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[27]  D. Dubois,et al.  Towards fuzzy differential calculus part 3: Differentiation , 1982 .

[28]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[29]  Juan J. Nieto,et al.  The Cauchy problem for continuous fuzzy differential equations , 1999, Fuzzy Sets Syst..