Neural profiles of cellular automata provide a systematic, computational framework for numer-ically and statistically characterizing seemingly disparate cellular automata transition rules. This computational framework utilizes graph convolution networks (GCNs) to exactly and precisely learn the full lookup table associated with a given cellular automaton rule. We address three hy- potheses all centered on addressing the precision and accuracy of intrinsically characterizing cellular automata transition rules independent of any extrinsic phenomenological behaviors conventionally studied in the literature. This thesis comprises of two main parts: the first part formally specifies the process of fitting graph convolutional networks and designing datasets of cellular automata for GCNs to learn; the second part then defines various methods for analyzing GCNs corresponding to a given cellular automaton rule.