An efficient deep neural network with amplifying sine unit for nonlinear oscillatory systems

Abstract

Nonlinear differential equations play a pivotal role in modeling various physical phenomena, capturing their oscillatory behaviors. Numerous analytic and numerical methods have been developed to obtain precise or approximate solutions for nonlinear dynamics. This study introduces a novel approach utilizing neural network strategies, known for their computational flexibility. This study proposed a novel architecture for deep neural networks that is intended to simulate the nonlinear oscillations of two systems. It has been modified to include an oscillatory activation function called the amplifying sine unit. This study compares the results from our customized neural networks with those from the conventional stable and reliable numerical technique of the Runge-Kutta method of order four. Remarkably, there is striking agreement between the outcomes of the two methods, highlighting the ability of deep neural networks to identify nonlinear dynamical behaviors without requiring explicit conversion of mathematical models into an equation system.