Approach to the cellular automaton interpretation with deep learning

In this paper, we will consider the machine learning system that can learn fundamental physics theory based on cellular automaton interpretation (CAI). First, assuming that we can calculate the time-evolved cellular automaton (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 certain example. Finding a CNN architecture that can learn CA is equivalent to showing that a time-varying time-evolution operator can be represented as a finite composition of time-independent linear functions and ReLU type non-linear functions. As a concrete example, block CA, which is time reversible and expressed as a matrix multiplication that changes with time, will be used as the time-evolution law, and the CNN architecture that can learn this evolution law will be proposed. However, by the universal approximation theorem, even with data of arbitrary quantum systems, if the depth of the network is deep enough, a CNN architecture that can learn actual rules can be found, regardless of the Hamiltonian, and therefore, the time-evolution law can be consistently expressed as a CNN. Also, since the convolution layer can be expressed in a covariant form, it could be helpful to find a CNN architecture that can learn the evolution law for a data set that includes gravity. Meanwhile, it will be shown that if the activation function of the first and last hidden layer is bypass, the CNN can be trained to include the corresponding part of the probabilistic interpretation in conventional quantum mechanics. Finally, for the CA model in which the dimensional reduction in quantum gravity is first presented, we will discuss the CNN architecture that can find the non-trivial evolution law in a deductive way.