Development a new conjugate gradient algorithm for improvement feedforward artificial neural networks for predicting the numbers of the Iraqi

a conjugate gradient method has been proposed for use in the training of Artificial Neural Networks with the feedforward technique. The proposed method demonstrated that the proposed CG method is accurate in predicting the numbers of the Iraqi population, the Our numerical results where very close to the exact solution of Tomas Malthose Model.


1.
Introduction As the human population is not a static factor, population growth is one of the key concerns of the world today. Instead, it is rising at an unprecedented pace. Despite the growing population of the planet, the resources of the earth remain constant. The need to maintain sustainable development is, thus becoming a big challenge for humanity today. All societies still need an accurate understanding of the future size of multiple entities for their future lives, such as population, wealth, demands, and consumption. Mathematical Modeling is a broad multidisciplinary science that uses mathematical and computational techniques to model and explains life science phenomena. A population model is a type of mathematical model that is applied to the study of population dynamics. In this study, we have modeled the population growth in Iraq using the differential equation according to Thomas's theory, as well as the use of artificial neural networks, and then compared the two methods.

2.
Thomas's theory [3][4] Thomas theory is based on the assumption that when population growth is not controlled, it increases according to a geometric progression. Where, during short periods of time, The number of births and deaths is proportional to the size of the population and the length of time. While the food increases according to the sequence of my account

Modeling the numbers of the population
Single population models using Thomas Malthus' theory is a matter of building a mathematical model of the population of a particular country or region by using differential equations to study the movement (dynamics) of living organisms.

Analysis of Population Growth in Mathematical Model of Thomas
According to Thomas Malthus' hypothesis that during time periods the numbers of births and deaths are proportional to both the size of society and the length of time model of population growth. It is based on the assumption that the population grows at a rate proportional to the size of the population. This is a reasonable assumption for a group of organisms under ideal conditions such as an indeterminate environment, adequate nutrition, and so on.

Mathematical formula
Assuming that: x N(t) Population variable with respect time t.
where it is assumed that "N(t)" is everywhere differentiable, the assumptions of continuity and differentiability provide reasonable approximations. [4] x ‫ݐߜ‬ Variable for a small-time specified.
x ‫ܣ‬ denotes the constant growth rate per capita of the population

Mathematical solution[1][2]
The Mathematical model for the numbers of the population is a first-order differential equation here ‫ܥ‬ is a constant. By solving this equation, we obtain the basic equation of Thomas's unconstrained growth model. It is stated that the size of the population would grow exponentially under these conditions by Equation (3) is the population exponential of the fundamental equation of unfettered growth models Thomas.
The population model is deterministic model, A Thomas model (3) depends on a known constant quantity ܰ(0) and an unknown constant "A".

Artificial Neural Networks:
Artificial neural networks (ANNs) are among the most powerful artificial intelligence techniques, as they are simulations of the biological neural network in the human brain. ANNs have many applications in different fields. One of the most amazing facts about ANNs is that they can compute any function at all.
In this section, we model the population growth of Iraq using ANNs ( Figure 2. for the structure of the ANN used in this paper). A new gradient-based conjugate CG method is proposed in order to train the proposed ANNs. The simulation outcomes show that the suggested techniques are very effective in predicting the population growth of Iraq.  [7][8] [9] In this section, we will show some of the conjugate gradient methods and then suggest a new algorithm for the conjugate gradient algorithm

Developed new CG algorithm
Where ߙ step-size that satisfy the standard Wolfe conditions or strong Wolfe conditions The Fletcher and Reeves (FR), Fletcher (CD), Polak and Ribiere (PRP) and ߚ is scalar.
Then the search direction Then from pure conjugacy condition

The Descent Property of a CG New Method
The descent property for our proposed new conjugate gradient scheme is shown below, denoted by ߚ ோௐ .

2-Let the relation ݃ ் ݀ < 0 for all ݇.
3-We're going to prove that the relation is true when ݇ = ݇ + 1 multiplying the equation in ݃ ାଵ we obtain

Global convergence study
We will display that CG method with ߚ ிோ and ‫ݐ‬ convergences globally. We need some assumptions for the convergence of the proposed new algorithm.

Lemma
Assume that Assumption (1) and equation (15) hold. take into consideration any conjugate gradient method in from (13) and (14)

Theorem (2)
Assume that Assumption (1) and equation (15) and the descent condition hold. Consider a conjugate gradient scheme in the form where ߙ is computed from strong Wolfe line search condition for more details see [17] [18], If the objective function is uniformly on set S, then ݈݅݉ →ஶ (݅݊ ‫ܧ‬ ∥ ݃ ∥) = 0. Proof Then lim →ஶ ݃ = 0

5-Numerical results of the developed ANNs algorithm
Results were shown by using simulation by PC-Laptob Dell win10 compatible with MATLAB (2018) using the core i7 processor with RAM 8GB and 7.89HDD. The results of the simulation artificial neural network were obtained by using the developed new CG algorithm training that illustrates the temporal behavior of population growth. It turns out that the network is effective when used to forecast in the short and long term because it is close to the real values of the population census for subsequent years table 3..  Fig(2) The results of simulated artificial neural network techniques are converging 6-Conclusions 1. The main results of this study are summarized in the following. 2. A mathematical model for population growth is constructed based on Thomas's theory. 3. ANNs have been employed to model and simulate population growth. Where a new gradient-based conjugate CG method is proposed in order to train the ANNs. 4. Simulation results showed that the ANNs are effective and accurate in predicting the population growth of Iraq. 5. The results of simulation of ANN prediction are better than Thomas' prediction model