Numerical solution of neutral delay differential equations using orthogonal neural network.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Artificial Intelligence

Background:

• Neutral delay differential equations (NDDEs) are crucial in modeling complex dynamic systems.
• Solving NDDEs, especially with variable coefficients and multiple delays, presents significant computational challenges.
• Existing methods often struggle with accuracy and efficiency for higher-order NDDEs.

Purpose of the Study:

• To introduce an efficient orthogonal neural network (ONN) approach for solving higher-order NDDEs.
• To demonstrate the method's capability in handling variable coefficients and multiple delays.
• To analyze the consistency and convergence properties of the proposed ONN method.

Main Methods:

• The core innovation involves replacing the feed-forward neural network's hidden layer with an orthogonal polynomial-based functional expansion block.
• Weights of the neural network are determined using the extreme learning machine (ELM) algorithm.
• The method is tested on various delay differential equations (DDEs) and NDDEs, including systems, using four types of orthogonal polynomials.

Main Results:

• The orthogonal neural network (ONN) method provides a uniform closed-form solution for NDDEs.
• The method achieves an error of order O(n^−m), where n is the number of neurons and m is related to the polynomial degree.
• Consistency and convergence analyses confirm the method's reliability.

Conclusions:

• The developed ONN approach offers an efficient and accurate technique for solving higher-order NDDEs.
• The method's flexibility with different orthogonal polynomials allows for tailored solutions.
• This research contributes a novel computational tool for problems involving time-delay systems.