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Product Code: 00007355

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The research work in this dissertation presents a new perspective for obtaining solutions of initial value problems using Artificial Neural Networks (ANN). We discover that neural network based model for the solution of Ordinary Differential Equations (ODEs) provides a number of advantages over standard numerical methods. First, the neural network based solution is differentiable and is in closed analytic form. On the other hand most other techniques offer a discretized solution or a solution with limited differentiability. Second, the neural network based method for solving a differential equation provides a solution with very good generalization properties. In our novel approach, we consider first, second and third order homogeneous and nonhomogeneous linear ordinary differential equations, and first order nonlinear ODE. In the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using SPSS statistical package. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis Function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD 14 software, which enables us to slightly adjust the intrinsic biases. For first, second and third order non-homogeneous ODE, we use the forcing function with the GRBF model to compute the weights from hidden layer to the output layer. The operational neural network model is redefined to incorporate the nonlinearity seen in nonlinear differential equations. We compare exact results with the neural network results for our example ODE problems. We find the results to be in good agreement. Furthermore these compare favourably with the existing neural network methods of solution. The major advantage here is that our method reduces considerably the computational tasks involved in weight updating, while maintaining satisfactory accuracy.



Title Page                                                                                                                    i

Certification                                                                                                                ii

Declaration                                                                                                                  iii

Dedication                                                                                                                  iv

Acknowledgement                                                                                                      v

Table of Contents                                                                                                       vi

List of Tables                                                                                                              viii

List of Figures                                                                                                             xi

Abstract                                                                                                                      xiii


CHAPTER 1:            INTRODUCTION                                                                            1

1.1 Definition of a Neural Network                                                                           1

1.2 Statement of the Problem                                                                                     3

1.3 Purpose of the Study                                                                                            4

1.4 Aim and Objectives                                                                                              4

1.5 Significance of the Study                                                                                     5

1.6 Justification of the Study                                                                                     6

1.7 Scope of the Study                                                                                               6

1.8 Definition of Terms                                                                                              6

1.9 Acronyms                                                                                                              7

CHAPTER 2:            REVIEW OF RELATED LITERATURE                                   9

CHAPTER 3:            MATERIALS AND METHODS                                        14

3.1 Artificial Neural Network                                                                                     14

3.1.1 Architecture                                                                                                       14

3.1.2 Training feed forward neural network                                                               15

3.2 Mathematical Model of Artificial Neural Network                                              15

3.3 Activation Function                                                                                              16

3.3.1 Linear activation function                                                                                  18

3.3.2 Sign activation function                                                                                     18

3.3.3 Sigmoid activation function                                                                              19

3.3.4 Step activation function                                                                                     19

3.4 Function Approximation                                                                                       19

3.5 General Formulation for Differential Equations                                                   21

3.6 Neural Network Training                                                                                      22

3.7 Method of Solving First Order Ordinary Differential Equations                         23

3.8 Computation of the Gradient                                                                               35

3.9 Regression Based Learning                                                                                  49

3.9.1 Linear regression: A simple learning algorithm                                      50

3.9.2 A neural network view of linear regression                                                       50

3.9.3 Least squares estimation of the parameters                                                       51


CHAPTER 4: RESULTS AND DISCUSSION                                                    53

4.1 First and Second Order Homogeneous Ordinary Differential Equation              53

4.2 First and Second Order Non-Homogeneous Ordinary Differential Equations    73

4.3 Third Order Homogeneous and Non-Homogeneous ODE                                  103

4.4 First and Second Order Linear ODE with Variable Coefficients:                       112

4.5 Nonlinear Ordinary Differential Equations (The Riccati Form of ODE)             122

4.6 Solving Nth order linear ordinary differential equations                                      131

4.7 Simulation                                                                                                             132

4.8 Discussion                                                                                                             144



5.1  Summary                                                                                                               145

5.2  Conclusion                                                                                                            146

5.3  Recommendations                                                                                                146

5.4  Contribution to knowledge                                                                                   146

                  References                                                                                                            147














LIST OF FIGURES                                           






A neural network is fundamentally a mathematical model, and its structure consists of a series of processing elements which are inter-connected and their operation resemble that of the human neurons. These processing elements are also known as units or nodes.  The ability of the network to process information is embedded in the connection strengths, simply called weights, which, when exposed to a set of training patterns, adapts to it. (Graupe 2007). The human brain consists of billions of nerve cells or neurons, as shown in Figure 1.1a. Neurons communicate through electrical signals which are short-lived impulses in the electromotive force of the cell wall. The neuron to neuron inter-connections are intermediated by electrochemical junctions called synapses, which are located on branches of the cell known as dendrite. Each neuron receives a good number of connections from other neurons, and there is constant incoming of multitude of signals, which each neuron receives and eventually gets to the cell body. Here, they are summed together in a way that if the resulting signal is greater than some threshold then the neuron will generate an impulse in response, coming from an electromotive force. This particular response is transmitted to other neurons through the axon which is a branching fibre. (Gurney, 1997). See figures 1.1a and 1.1bs.

Neural network methods can solve both ordinary as well as partial differential equations. And it relies on the function approximation seen in feed- forward neural networks which results in a solution written in an analytic form. This form employs a feed forward neural network as a basic approximation element. (Principe et al., 1997). Training of the neural network can be done either by any optimization technique which in turn requires the computation of the derivative of the error with respect to the network parameters, by regression based model or by basis function approximation. In any of these methods, a neural network solution of the given differential equation is assumed and designated a trial solution which is written as a sum of two parts, proposed by Lagaris et al., (1997). The first part of the trial solution satisfies the conditions prescribed at the initial or boundary, and contains non-of the parameters that need adjustment. The other part contains some adjustable parameters that involves feed- forward neural network and is constructed in a way that does not affect the conditions. Through the construction, the trial solution, initial or boundary conditions are satisfied and the network is trained to satisfy the differential equation.


 Fig.1.1 a            Biological Neuron (Carlos G., Online)


Teaching Input


Figure 1.1 b An Artificial Neuron (Yadav et al., 2015)

It is this architecture in Figure 1.1b, and style of processing that we hope to incorporate in neural networks solution of differential equations.



In this research, we propose a new method of solving ordinary differential equations (ODEs) with initial conditions through Artificial Neural Network (ANN) based models. The conventional way of solving differential equations using artificial neural network involves updating of all the parameters, weights and biases, during the neural network training. This is caused by the inability of the neural network to predict a solution with an acceptable minimum error. In order to reduce the error, the error function is minimized. Minimizing the error function demands finding its gradient. This gradient involves the computation of multivariate partial derivatives of the error function with respect to all the parameters, weights and biases, and the independent variable. This is quite involving as we shall demonstrate later for first order differential equation. It is even more difficult when solving second or higher order ODE where you need to find the second or higher order derivative of the error function. This research work involves systematically computing the weights such that no updating is required, thereby eliminating the herculean task in finding the partial derivative of the error function.



The main purpose of embarking on this research is to explore an avenue or an approach to reducing the herculean task involved in weight updating in the process of neural network training of the parameters associated with the minimization of the error function, which in turn involves multivariate partial derivatives with respect to all the parameters and the independent variables.


Aim:    The aim of this work is to solve both linear and nonlinear ordinary differential equations using Artificial Neural Network (ANN) model, by implementing the new approach which this study proposes. We shall achieve the aim through the following objectives:

Objectives: We shall   systematically

(i)                 compute the weights from input layer to hidden layer using regression based model

(ii)               compute the weights from hidden layer to output layer using Radial Basis Function (RBF) model

(iii)             slightly adjust the biases using Mathematical Computer Aided Design (MathCAD) 14 software algorithm to achieve the desired accuracy

(iv)             develop a neural network that will incorporate the nonlinearity found in such ODE as the Riccati type

(v)               Suggests a way of tackling nth order ODE

(vi)             Compare our results with analytical results and some other neural network result.

(vii)           Simulate our results to show how they agree with other solutions



A neural network based model for solving differential equations provides the following advantages over the standard numerical methods:

a.                   The neural network based solution of a differential equation is differentiable and is in closed analytic form that can be applied in any further calculation. On the other hand most other methods like Euler, Runge-Kutta, finite difference, etc, give a discrete solution or a solution that has limited differentiability.

b.                  The neural network based method for solving a differential equation makes available a solution with fantastic generalization properties.

c.                   Computational complexity does not increase rapidly in the neural network method when the number of points to be sampled is increased while in the other standard numerical methods computational complexity increases rapidly as we increase the number of sampling points in the interval. Most other approximation methods are observed to be iterative in nature, and the step size fixed before the beginning of the computation. ANN offers some reliefs to overcoming some of these repetition of iterations. Now, after the ANN has converged, we may use it as a black box to get numerical results at any randomly picked points in the domain.

d.                  The method is general and can be applied to the systems defined on either orthogonal box boundaries or on irregular arbitrary shaped boundaries.

e.                   Models based on neural network offers an opportunity to handle difficult differential equation problems arising in many sciences and engineering applications.

f.                   The method can be implemented on parallel architectures. (Yadav et al. 2015)


The new approach we are proposing in this research will eliminate the computation of partial derivative of the error function thereby reducing the task involved in using neural network to solve differential equations.


This study covers first, second and third order linear and first order nonlinear ODE with constant and variable coefficients. It is also extended to the nth order linear ODE, all with initial conditions. It does not include ODE with boundary conditions, other nonlinear ODE with the product of the dependent variable and its derivative, and partial differential equations.



Nodes: are computational units which receive inputs, and process them into output.

Synapses: are connections between neurons. They determine the information flow which exists between nodes.

Weights: are the respective signaling strength. The ability of the network to process information is stored in connection strength, simply called weights.

Neurons: are the primary signaling units of the central nervous system and each neuron is a distinct cell whose several processes arise from its cell body. A neuron is the basic processor or processing element in a neural network. Each neuron receives one or more input over its connections and produces only one output.

Architecture: is the pattern of connections between the neurons which can be a multilayer feed forward neural network architecture. (Tawfiq & Oraibi, 2013). When a neural network is in layers, the neurons are arranged in the form of layers. There are a minimum of two layers: an input layer and an output layer. The layers between the input layer and the output layer, if they exist, are referred to as hidden layers, and their computation nodes are referred to as hidden neurons or hidden units. Extra neurons at the hidden layers raise the network’s ability to extract higher-order statistics from (input) data. (Alaa, 2010).

Training: is the process of setting the weights and biases from the network, for the desired output.

Regression: is a least-squares curve that fits a particular data.

Goodness of Fit (R2): is a terminology used in regression analysis to tell us how good a given data has fit the regression model

Neural Network: is interconnection of processing elements, which resemble that of human neurons.

Artificial Neural Network: is a simplified mathematical model of human brain, also known as information processing system.

Activation function: is a threshold or transfer function (non-linear operator), which keeps the cell’s output between certain limits as is the case in the biological neuron.

Axon: conducts electric signals down its length.

Bias: is a parameter which helps to speed up convergence. The addition of biases increases the flexibility of the model to feed the given data. Bias determines if a neuron is activated. The performance of an activation function ought to be propagated forward through the network. The bias term in the network determines whether or not this will happen. The absence of bias hinders this forward propagation, leading to undesirable outcome.


1.9       ACRONYMS:

ANN – Artificial Neural Network

BVP – Boundary Value Problem

CPROP - Constrained-Backpropagation

FFNN – Feed Forward Neural Network

GRBF - Gaussian Radial Basis function

IVP – Initial Value Problem

MathCAD – Mathematical Computer Aided Design

MLP – Multi Layer Perceptron

MSE – Mean Squared Error

NN – Neural Network

ODE – Ordinary Differential Equation

PDE – Partial Differential Equation

PDP - Parallel Distributed Processing                                            

PE - Processing Elements

RBA – Regression Based Algorithm

RBF – Radial Basis Function

RBFNN – Radial Basis Function Neural Network

SPSS – Statistical Package for Social Sciences




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