Narrow Results

The structure of a novel computational hyperbolic tangent sigmoid deep neural network (HTS-DNN) is presented for the numerical solutions of the hepatitis B virus model, which is based on the antibody immune response. The mathematical model is categorized as healthy and hepatocytes, capsids, antibodies and free viruses. A novel process based on the HTS-DNN is exploited by using two hidden layers with 20 and 30 numbers of neurons. The optimization is performed through Bayesian regularization, which is one of the reliable procedures used in the optimization of various problems. A dataset is obtained through the Runge–Kutta solver, which is used to reduce the mean square error by dividing the training, testing and verification data as 70%, 16% and 14%. Moreover, the statistical representations in the sense of error histogram, regression, and state transitions also approve the accuracy of the scheme.

The “hard water” factor shows the management of water in the Nusa Tenggara Timur, which shows a higher ratio based on the ion’s minerals. The incessant use of hard water presents kidney dysfunction, which produces diabetic and vascular kinds of diseases. Therefore, it is essential to recognize the influences of hard water on kidney function. A novel design of a stochastic solver using the transfer radial basis function is provided by applying the Bayesian regularization neural network for solving the model. The kidney dysfunction mathematical system is divided into humans (susceptible, infected, recovered) and water components (magnesium, calcium). Twelve numbers of neurons with the radial basis transfer function have been used in the hidden layers for solving the model. The approach performance is remarked through the results comparison and further reducible absolute error found around $10^{-06}$ to $10^{-08}$ develop the scheme’s exactness. Moreover, the statistical performances including regression coefficient performances around 1 for each case of the model validate the reliability and exactness of the scheme for solving the model.

In this paper, a new local RBF collocation approach based on point interpolation method, which can assure that its coefficient matrices are of bandwidth, has been presented. The main feature of this approach is to use a Hermite-type interpolation scheme, namely making use of the normal gradient at Neumann boundary so that this computational accuracy has greatly improved. The accuracy and simplicity of this presented approach will be shown numerically by well-known 2-D linear elastic Cook's membrane benchmark.

Evolutionary algorithms (EAs) are population based approaches that start with an initial population of candidate solutions and evolve them over a number of generations to finally arrive at a set of desired solutions. Such population based algorithms are particularly attractive for multi-objective optimization (MOO) problems as they can result in a set of non-dominated solutions in a single run. However, they are known to require evaluations of a large number of candidate solutions during the process that often becomes prohibitive for problems involving computationally expensive analyses. Use of multiple processors and cheaper approximations (surrogates or metamodels) in lieu of the actual analyses are attractive means to contain the computational time within affordable limits. A major problem in using surrogates within an evolutionary algorithm lies with its representation accuracy; the problem is far more acute for multi-objective problems where both the proximity to the Pareto front and the diversity of the solutions along the non-dominated front are required. In this chapter, a surrogate assisted evolutionary algorithm (SAEA) for multi-objective optimization is presented.

A Radial Basis Function (RBF) network is used as a surrogate model. The algorithm performs actual evaluations of objectives and constraints for all the members of the initial population and periodically evaluates all the members of the population after every S generations. An external archive of the unique solutions evaluated using actual analysis is maintained to train the RBF model which is then used in lieu of the actual analysis for the next S generations. In order to ensure the prediction accuracy of the RBF surrogate model, a candidate solution is only approximated if at least one candidate solution in the archive exists in the vicinity (based on a user defined distance threshold) and the accuracy of the surrogate is within an user defined limit. Five multi-objective test problems are presented in this study and a comparison with Nondominated Sorting Genetic Algorithm II (NSGA-II) [Deb et al. (2002a)] is included to highlight the benefits offered by the approach. SAEA algorithm consistently reported better nondominated solutions for all the test cases for the same number of actual evaluations of candidate solutions.