Artificial Neural Networks to Solve the Singular Model with Neumann-Robin, Dirichlet and Neumann Boundary Conditions

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Soft Computing

Background:

• Singular models with mixed boundary conditions (Neumann-Robin, Dirichlet, Neumann) present significant computational challenges.
• Existing numerical methods may struggle with the robustness and stability required for these complex systems.

Purpose of the Study:

• To develop and validate a novel hybrid computational framework, ANN-GA-SQPM, for solving singular models.
• To assess the robustness, stability, and proficiency of the proposed ANN-GA-SQPM method.
• To compare the accuracy of the ANN-GA-SQPM results against exact solutions.

Main Methods:

• A hybrid computational framework combining artificial neural networks (ANNs), a global search genetic algorithm (GA), and a local search sequential quadratic programming method (SQPM) was developed.
• The ANN-GA-SQPM framework was applied to four distinct singular problems featuring Neumann-Robin, Dirichlet, and Neumann boundary conditions.
• Statistical performance analysis and neuron count variations (3 vs. 15 neurons) were employed to validate the method's authenticity.

Main Results:

• The ANN-GA-SQPM framework successfully solved the singular models across all tested boundary conditions.
• Comparative analysis with exact solutions confirmed the high accuracy and efficiency of the proposed method.
• Neuron analysis indicated the reliability and authenticity of the ANN-GA-SQPM approach.

Conclusions:

• The developed ANN-GA-SQPM framework offers a reliable and proficient computational tool for addressing singular models with mixed boundary conditions.
• The hybrid approach effectively integrates the strengths of ANNs and soft computing optimization techniques.
• This study validates the ANN-GA-SQPM method's potential for solving complex mathematical problems in various scientific domains.