Supervised and unsupervised learning of directed percolation

Machine learning (ML) has been well applied to studying equilibrium phase transition models by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into nonequilibrium phase transitions. The extra dimension in a given nonequilibrium system, namely time, can greatly slow down the procedure toward the steady state. In this paper we find that by using some simple techniques of ML, non-steady-state configurations of directed percolation (DP) suffice to capture its essential critical behaviors in both (1+1) and (2+1) dimensions. With the supervised learning method, the framework of our binary classification neural networks can identify the phase transition threshold, as well as the spatial and temporal correlation exponents. The characteristic time t_{c}, specifying the transition from active phases to absorbing ones, is also a major product of the learning. Moreover, we employ the convolutional autoencoder, an unsupervised learning technique, to extract dimensionality reduction representations and cluster configurations of (1+1) bond DP. It is quite appealing that such a method can yield a reasonable estimation of the critical point.