Physics-based Machine Learning for quantifying chaos in the gravitational three-body problem

The three-body problem consists of determining the motion of three masses under their mutual gravitational attraction, leading to unpredictable trajectories due to non-linear interactions. It inherently exhibits chaotic behavior due to its sensitivity to initial conditions, challenging both analytical and numerical approaches. This thesis aims to leverage Machine Learning (ML), specifically feedforward neural networks, to discern the chaos within such systems. To do so, we measure the rate at which nearby trajectories in phase space diverge over time, by means of Lyapunov exponents. They can be approximated by computing the slope of the logarithm of the separation between the two solutions, over a finite-time interval. The inverse of the largest Lyapunov exponent defines the Lyapunov timescale, which characterizes how quickly the system's behavior becomes unpredictable. The objective is to predict the main features of the distribution of local Lyapunov exponents, computed over the entire trajectory of the triple. However, thanks to the ML network, the information about chaos can be extracted without the need for a full numerical integration, starting from a single set of coordinates in the phase space. In this way, we avoid computational expenses and get a quick snapshot of the intrinsic chaos of a specific system. This approach aspires to advance our understanding of chaotic behavior in the three-body problem and to contribute novel methodologies to the broader field of dynamical systems analysis.