Approach to the cellular automaton interpretation with deep learning

In this paper, we will consider the deep learning systems that can learn fundamental physics theory based on cellular automaton interpretation (CAI). First, assuming that we can map quantum states to cellular automaton (CA) and calculate the time-evolved CA for any initial CA by knowing the time-evolution law of the given system, we will show that there exists a convolutional neural network (CNN) architecture that can learn the time-evolution law of this system with only the calculated data set for a time-reversible CA. Mathematically, finding a CNN architecture that can learn CA rule is equivalent to showing that a time-evolution operator can be approximated as a finite composition of time-independent linear functions and ReLU type non-linear functions, as the possible associated generator of approximation may absorbs the information about the dynamics. Going one step further, we will discuss the correspondence between the quantum system and deep learning architecture and relate the concept of moduli space of Riemann surfaces to deep learning parameters when considering interactions. Finally, for the CA model in which the dimensional reduction in quantum gravity was first presented, we will discuss the CNN architecture that can find the non-trivial evolution law for holographic direction in a deductive way without the label. It is suggested that the limits to this effort can be improved through AdS/CFT correspondance.