Design of Mayer Wavelet Neural Networks for Solving Functional Nonlinear Singular Differential Equation

Table 1

Pseudocode of GASQP optimization process for FM-ANN.

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Genetic algorithms started

 Inputs:

  The chromosome having equal entries is unknown in FM-ANN as

  , for

  Initial population:

  , for

 Output:

  The optimize best weights for FM-ANN by GA, W_[Best-GA].

 Initialization:

  Construct W with real entries and set of W to form P for the “GA” and “gaoptimset” routines

 Fitness formulation:

  Obtain the value of the E for each W in P by equation (5)

 Termination:

  Stop the algorithm, when meet any of the following

  Fit = E⟶10^-16,

  PopulationSize⟶ 200,

  TolCon⟶10^-22,

  Tolerances = TolFun⟶10^-18,

  StallGenLimit⟶110,

  Generations⟶70 and

  other default values

  when termination condition meets, move to storage step

 Ranking:

  Rank each W of P on E on the basis of equation (5)

 Reproduction:

  Create new P with the use of “Selection,” “Crossover,” and “Mutations” routines. For best ranked W of P for elitism,

  Go to the step of “fitness evaluation”

 Storage:

  Save W_[Best-GA], E, time, generation and counts of function

End of genetic algorithms

Start the procedure of SQ programming

 Inputs:

   W_[Best-GA]: GA initial weights,

 Output:

  The best weights for FM-ANN by GASQP scheme, W_GASQ

 Initialize:

  Start point of the algorithm is W_[Best-GA]

  Set constraints limits, iterations/generations, bounded and other

 Terminate:

  Stop if the following criteria meets

  Fitness = E ≤ 10^-14,

  iterations = 600,

  tolerance; TolFun ≤ 10^-18, TolX ≤ 10^-22, TolCon ≤ 10^-18 MaxFunEvals ≤ 220000, and

  other values are default

 While (terminate)

  Fitness formulation:

  Calculate E by equation (5)

 Fine tuning:

  Operate “fmincon” routine with ‘sequential quadratic programming’ for quick adjustment of W at each stage

  Go to fitness evaluation step with upgraded W

 End

 Accumulate:

  Store the W_GASQ, E, time, iterations and function counts

SQ programming process End

Dataset generation:

Repeat 100 times the GASQP procedure to make a dataset of the optimization variables for FM-ANN to solve singular NFDE system for effective statistical implications

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