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

Susmita Mall1, S Chakraverty2

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A novel Hermite neural network (HeNN) effectively solves nonlinear oscillator equations, offering reliable approximate solutions for complex problems like early mechanical failure detection.

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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.