Artificial neural network approach for solving fuzzy differential equations

Abstract

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.

Introduction

Uncertainty is an attribute of information, [28] and the use of fuzzy differential equations (FDEs) is a natural way to model dynamic systems with embedded uncertainty. Most practical problems can be modeled as FDEs (e.g. [5], [8] and Section 3.2). The method of fuzzy mapping was initially introduced by Chang and Zadeh [10]. Later, Dubois and Prade [11] presented a form of elementary fuzzy calculus based on the extension principle [27]. Puri and Ralescue [23] suggested two definitions for the fuzzy derivative of fuzzy functions. The first method was based on H-difference notation and was further investigated by Kaleva [16]. Several approaches were later proposed for FDEs and the existence of their solutions (e.g. [15], [19], [21], [24], [26]). The approach based on H-derivative has the disadvantage that it leads to solutions which have an increasing length of their support. This shortcoming was resolved by interpreting the FDE as a family of differential inclusions. Later, the authors of [6], [7] introduced the concept of generalized differentiability. According to this new definition, the solution of the FDE may have decreasing length of its support. Other researchers have proposed several approaches to the solutions of FDE (e.g. [9], [19]).

Another group of researchers tried to extend some numerical methods to solve FDEs (e.g. [1], [12], [13]) such as Runge–Kutta method [2], Adomian method [4], predictor–corrector method and multi-step methods [3]. These methods are extended versions of the equivalent methods for solving ordinary differential equations (ODEs).

Lagaris et al. [17] used artificial neural networks to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) for both boundary value problems and initial value problems. They used multilayer perceptron to estimate the solution of differential equation. Their neural network model was trained over an interval (over which the differential equation must be solved), so the inputs of the neural network model were the training points. The comparison of their method with the existing numerical methods shows that their method was more accurate and the solution had also more generalizations. The ability of neural networks in function approximation is our main objective. In this paper, we will construct a new model with the use of neural networks to obtain a solution for FDE. In this new model, the inputs of the neural network are the training points as well as a parameter r which shows the domain of uncertainty.

In Section 2, the basic notations of fuzzy numbers, fuzzy derivative and fuzzy functions are briefly presented. In Section 3, fuzzy differential equations and their applications are introduced; in addition, a general Cauchy problem is defined. In Section 4, the proposed new method (based on the use of a feed-forward neural network) is described. In Section 5, the applicability of the method is illustrated by several examples in which the exact solution and the computed results are compared with each other. In order to show the applicability of the method, a nonlinear FDE and a nonlinear FDE containing a fuzzy parameter are solved. Also, a problem in electrical circuit analysis is solved. Finally, Section 6 presents concluding remarks.