Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation

Area of Science:

• Computational mathematics
• Artificial intelligence
• Nonlinear dynamics

Background:

• Nonlinear oscillator equations, such as the Van der Pol-Duffing equation, often lack exact analytical solutions.
• Approximate numerical methods are crucial for understanding and predicting the behavior of these systems.

Purpose of the Study:

• To introduce a novel Hermite polynomial-based functional link artificial neural network (HeNN) for solving nonlinear oscillator equations.
• To demonstrate the HeNN model's capability in providing accurate approximate solutions for the first time.

Main Methods:

• A single-layer Hermite neural network (HeNN) was developed, utilizing Hermite orthogonal polynomials for input pattern expansion.
• A feedforward neural network architecture with unsupervised error backpropagation was employed to train the HeNN model.
• The HeNN model was applied to mathematical examples and real-world problems, including mechanical failure signal analysis and weak signal detection.

Main Results:

• The HeNN model successfully generated approximate solutions for the Van der Pol-Duffing and Duffing oscillator equations.
• HeNN solutions were validated against the established Runge-Kutta method, showing comparable accuracy.
• The trained HeNN model functions as a reliable black box for obtaining numerical results at arbitrary points.

Conclusions:

• The proposed Hermite neural network (HeNN) method provides an efficient and reliable approach for solving nonlinear oscillator equations.
• This novel method demonstrates significant potential for application to a wide range of other nonlinear problems.
• The HeNN model offers a powerful tool for analyzing complex systems in engineering and scientific research.