Supervised and Unsupervised Learning with Numerical Computation for the Wolfram Cellular Automata

The local rules of elementary cellular automata (ECA) with one-dimensional three-cell neighborhoods are represented by eight-bit binary numbers that encode deterministic update rules. This class of systems is also commonly referred to as the Wolfram cellular automata. These automata are widely utilized to investigate self-organization phenomena and the dynamics of complex systems. In this work, we employ numerical simulations and computational methods to investigate the asymptotic density and dynamical evolution mechanisms in Wolfram automata. We explore alternative initial conditions under which certain Wolfram rules generate similar fractal patterns over time, even when starting from a single active site. Our results reveal the relationship between the asymptotic density and the initial density of selected rules. Furthermore, we apply both supervised and unsupervised learning methods to identify the configurations associated with different Wolfram rules. The supervised learning methods effectively identify the configurations of various Wolfram rules, while unsupervised methods like principal component analysis and autoencoders can approximately cluster configurations of different Wolfram rules into distinct groups, yielding results that align well with simulated density outputs. Machine learning methods offer significant advantages in identifying different Wolfram rules, as they can effectively distinguish highly similar configurations that are challenging to differentiate manually.