Modeling chaotic Lorenz ODE System using Scientific Machine Learning

The Lorenz system of equations is a set of ordinary differential equations to represent a simplified model of atmospheric convection Sparrow [1982]. These set of equations have a wide range of applications in fields ranging from fluid mechanics to laser physics to weather prediction. One of the most interesting properties of the Lorenz ODE System is that it is chaotic in nature Fowler et al. [1982]. Small changes in the initial conditions can lead to vastly different outcomes in the end result Liao S. [2014]. When simulated over a given period, the Lorenz ODEs show oscillations in time. Usually, numerical methods implemented in computational software modeling tools like Python, Julia, or Matlab are used to simulate the Lorenz System of ODEs. These methods are inefficient as Lorentz equations are sensitive to initial conditions and minute changes to the conditions and tiny rounding errors can lead to the accumulation of numerical errors over time. Very few studies have been aimed at integrating machine learning-aided methods in simulating the chaotic Lorenz system. In this study, we provide a robust investigation of the effect of two physics-aided machine learning models in simulating the Lorenz system of ODEs: Neural Ordinary Differential Equations (Neural ODEs) Chen et al. [2018] and Universal Differential Equations (UDEs) Rackauckas et al. [2020a].