Oscillator Simulation with Deep Neural Networks

Jamshaid Ul Rahman, Sana Danish and Dianchen Lu ()
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Jamshaid Ul Rahman: School of Mathematical Sciences, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China
Sana Danish: Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
Dianchen Lu: School of Mathematical Sciences, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China

Mathematics, 2024, vol. 12, issue 7, 1-15

Abstract: The motivation behind this study is to overcome the complex mathematical formulation and time-consuming nature of traditional numerical methods used in solving differential equations. It seeks an alternative approach for more efficient and simplified solutions. A Deep Neural Network (DNN) is utilized to understand the intricate correlations between the oscillator’s variables and to precisely capture their dynamics by being trained on a dataset of known oscillator behaviors. In this work, we discuss the main challenge of predicting the behavior of oscillators without depending on complex strategies or time-consuming simulations. The present work proposes a favorable modified form of neural structure to improve the strategy for simulating linear and nonlinear harmonic oscillators from mechanical systems by formulating an ANN as a DNN via an appropriate oscillating activation function. The proposed methodology provides the solutions of linear and nonlinear differential equations (DEs) in differentiable form and is a more accurate approximation as compared to the traditional numerical method. The Van der Pol equation with parametric damping and the Mathieu equation are adopted as illustrations. Experimental analysis shows that our proposed scheme outperforms other numerical methods in terms of accuracy and computational cost. We provide a comparative analysis of the outcomes obtained through our proposed approach and those derived from the LSODA algorithm, utilizing numerical techniques, Adams–Bashforth, and the Backward Differentiation Formula (BDF). The results of this research provide insightful information for engineering applications, facilitating improvements in energy efficiency, and scientific innovation.

Keywords: deep neural network; harmonic oscillator; Mathieu equation; nonlinear dynamics; Van der Pol equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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