Cellular automata as convolutional neural networks

Deep-learning techniques have recently demonstrated broad success in predicting complex dynamical systems ranging from turbulence to human speech, motivating broader questions about how neural networks encode and represent dynamical rules. We explore this problem in the context of cellular automata (CA), simple dynamical systems that are intrinsically discrete and thus difficult to analyze using standard tools from dynamical systems theory. We show that any CA may readily be represented using a convolutional neural network with a network-in-network architecture. This motivates the development of a general convolutional multilayer perceptron architecture, which we find can learn the dynamical rules for arbitrary CA when given videos of the CA as training data. In the limit of large network widths, we find that training dynamics are nearly identical across replicates, and that common patterns emerge in the structure of networks trained on different CA rulesets. We train ensembles of networks on randomly sampled CA, and we probe how the trained networks internally represent the CA rules using an information-theoretic technique based on distributions of layer activation patterns. We find that CA with simpler rule tables produce trained networks with hierarchical structure and layer specialization, while more complex CA produce shallower representations—illustrating how the underlying complexity of the CA's rules influences the specificity of these internal representations. Our results suggest how the entropy of a physical process can affect its representation when learned by neural networks.

Figure 1

Conway's Game of Life as a convolutional neural network. Two convolutional filters identify the value of the center pixel and count the number of neighbors. These features are then scored and summed to generate a prediction for the system at the next timepoint.

Figure 2

Architecture of a trainable convolutional neural network for learning cellular automata. Dimensions, where not marked, are determined by the dimensionality of the previous layer.

Figure 3

Training 2560 convolutional neural networks on random cellular automata. (a) A network trained on the Game of Life for different durations, and then applied to images of each stage of the “glider” solution. (b) The loss versus time during training, colored by the rule entropy ${\mathcal{H}}_{\mathrm{ca}}$. Groups of 512 related cellular automata were generated by iteratively choosing random $\sigma \to 0$ rules from the 512 possible input configurations, and setting those sites to $\sigma \to 1$. Five replicates were performed. Loss values represent the sum over the batch; values of 10 or smaller imply that only small rounding errors were present at the end of training. The entropy of the resulting rule table is characteristic of the CA, and it is indicated by ${\mathcal{H}}_{\mathrm{ca}}=0$ (blue, minimum entropy CA) to ${\mathcal{H}}_{\mathrm{ca}}=9$ (magenta, maximum entropy CA). (Inset) The final loss for each network at the end of training, shown as a function of ${\mathcal{H}}_{\mathrm{ca}}$.

Figure 4

Internal representations of cellular automata by trained networks. (a) The individual layerwise entropy (${\mathcal{H}}_{\mathsf{L},i}/D$) for the 2560 networks shown in the previous figure. Noise has been added to the horizontal coordinates (layer index) to facilitate visualization. As in previous figures, coloration corresponds to the entropy ${\mathcal{H}}_{\mathrm{ca}}$ of the underlying CA. Dashed lines correspond to expected trends for theoretical networks that eliminates $0\%$ of cases in each layer (i.e., a pattern-matching implementation), $45\%$ of cases, and $50\%$ (top to bottom) (Inset) The Pearson correlation coefficient $r$ between the rule entropy ${\mathcal{H}}_{\mathrm{ca}}$ and layer entropy ${\mathcal{H}}_{\mathsf{L},i}$. Error range corresponds to bootstrapped 25%–75% quantiles. (b) The normalized layerwise entropy (${\mathcal{H}}_{\mathsf{L},i}/D$) versus the normalized total layerwise neuron entropy (${\mathcal{H}}_{\mathsf{N},ij}/N$), with the linear scaling annotated.