Learnable and time-reversible cellular automata with holographic principle

Recently, there are active studies to extend the concept of convolutional neural networks(CNNs) to non-Euclidean space. In particular, there have been a study on how to implement CNNs for data in non-Euclidean space that are invariant under a certain transformation. During this process, the concept of symmetry came in and convolution was described as a covariant form that the physics theory should be satisfied with after considering gauge symmetry. However, just because the convoultion expressed in covariant form is obtained, it is not obvious to implement the algorithm corresponding to that expression. Here, the universal approximation theorem tells us that any function can be approximated to a feed-forward networks. Therefore, the already known mathematical expression of covariant CNNs can be implemented through feed-forward neural networks. In this point of view, we demonstrate to learning process of cellular automata(CA) that could satisfy locality,time-reversibility and the certain holographic principle through conventional CNNs. With simple rules that satisfy the above three conditions and an arbitrary dataset that satisfies those rules, CNNs architecture that can learn rules were proposed and it was confirmed that accurate inferences were made for simple examples.