PINNs Study for the Bekki-Nozaki Chaos in the Non-linear Schrödinger equation

Abstract: In this paper we study chaotic behavior in the forced dissipative non-linear Schr\"{o}dinger equation, so called the Bekki-Nozaki equation. Chaotic systems are often seen in a strong sensitivity to initial conditions,leading to error accumulation over time when traditional numerical methods are applied. To address this difficulty, we employ Physics-Informed Neural Networks(PINNs), a mesh-free deep learning framework. PINNs mitigate error accumulation in chaotic systems by solving partial differential equations without discretizing the computational domain. We demonstrate that PINNs successfully reproduce chaotic behavior of the Bekki-Nozaki equation. The results of the inverse analysis indicate a correlation between the governing equation's predictability and its chaotic nature of the solution.