5. Fitness Evaluation
Calculate the fitness f(i) for every chromosome as:
where M denotes the number of input patterns in train data, x(j) is the j train pattern and t(j) is the desired output.
We express the ODE's in the following form:
where y^(n) denotes the n-order derivative of y. Let the boundary or initial conditions be given by:
where t(i) is one of the two endpoints a or b. The steps for the fitness evaluation of the population are the following:
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Choose N equidistant points (x(0),x(1),...,x(N-1)) in the relevant range.
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For every chromosome i
(a) Construct the corresponding model M(i)(x), expressed in the grammar described in section 1.
(b) Calculate the quantity
(c) Calculate an associated penalty P(M_{i}) as shown below.
(d) Calculate the fitness value of the chromosome as: 
The penalty function P depends on the boundary conditions and it has the form:
where λ is a positive number.
The proposed method can solve systems of ordinary differential equations that are expressed in the form:
with initial conditions:
The steps for the fitness evaluation are the following:
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Choose N equidistant points (x(0),x(1),...,x(N-1)) in the relevant range.
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For every chromosome i
(a) Split the chromosome uniformly in k parts, where k is the number of equations in the system.
(b) Construct the k models M(i),j=1..k
(c) Calculate the quantities
for j=1..k
(d) Calculate the quantity
(e) Calculate the associated penalties
for j=1...k where λ is a positive number.
(f) Calculate the total penalty value
(g) Finally, the fitness of the chromosome i is given by:
We only consider here elliptic PDE's in two and three variables with Dirichlet boundary conditions. The generalization of the process to other types of boundary conditions and higher dimensions is straightforward. The PDE is expressed in the form:
with x in [x(0),x(1)] and y in [y(0),y(1)]. The associated Dirichlet boundary conditions are expressed as:
The steps for the fitness evaluation of the population are given below:
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Choose N^2 equidistant points in the box [x(0),x(1)] X [y(0),y(1)], N(x) equidistant points on the boundary at x=x(0) and at x=x(1), N(y) equidistant points on the boundary at y=y(0) and at y=y(1).
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For every chromosome i
• Construct a trial solution M(i)(x,y) expressed in the grammar described earlier.
• Calculate the quantity
• Calculate the quantities
• Calculate the fitness of the chromosome as:
