Machine Learning Of Phase Transitions In Unitary And Binary Reaction-diffusion Processes

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 non-equilibrium phase transitions.The extra dimension in a given non-equilibrium system,namely time,can greatly slow down the procedure towards the steady state.In this thesis,we find that by using some techniques of ML,non-steady state configurations of directed percolation(DP)and pair contact process with diffusion(PCPD)suffice to capture its essential critical behaviors.DP is a unitary random process,while PCPD is a binary reaction-diffusion process,which includes a diffusion of particles.There are still many unsolved problems in non-equilibrium phase transitions,such as problems of analytical solutions and universality class.The DP universality class is the most important known one in non-equilibrium phase transitions,but its(1+1)dimension analytical solution has not been obtained up to now.Taking PCPD model as an example,its universality class is still a very controversial topic,and there is no conclusion at present.Recently,with the wide application of ML techniques in statistical physics,we have focused our attention on the study of non-equilibrium phase transitions by ML.When studying phase transition configurations,ML is suitable for small size cases.With the supervised learning method,the framework of our binary classification neural networks can identify the phase transition threshold and the spatial and temporal correlation exponents of the DP model.The characteristic time tc,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.The PCPD,a generalized model of the ordinary pair-contact process(PCP)without diffusion,exhibits a continuous absorbing phase transition.Unlike the PCP,whose nature of phase transition is clearly classified into the DP universality class,the model of PCPD has been controversially discussed since its infancy.To our best knowledge,there is so far no consensus on whether the phase transition of the PCPD falls into the unknown university classes or else conveys a new kind of non-equilibrium phase transition.In this thesis,both unsupervised and supervised learning are employed to study the PCPD with scrutiny.Firstly,two unsupervised learning methods,principal component analysis(PCA)and autoencoder,are taken.Our results show that both methods can cluster the original configurations of the model and provide reasonable estimates of thresholds.Therefore,no matter whether the non-equilibrium lattice model is a random process of unitary(for instance the DP)or binary(for instance the PCP),or whether it contains the diffusion motion of particles,unsupervised leaning can capture the essential,hidden information.Beyond that,supervised learning is also applied to learning the PCPD at different diffusion rates.We proposed a more accurate numerical method to determine the spatial correlation exponent v⊥,which,to a large degree,avoids the uncertainty of data collapses through naked eyes.Our extensive calculations reveal that the Monte Carlo(MC)simulations of small-sized systems showed larger deviations compared to the counterparts of large-sized systems.These deviations are caused partly by the randomness of diffusion,and partly by the finite sizes.But supervised ML results are revealed to be relatively stable and much less affected by the finite sizes.We believe that supervised ML to calculate the critical exponents can provide some reference for the study of the PCPD.In addition,we also apply an unsupervised learning method named the autoencoder neural network to extract critical points of phase transitions.Phase transition models involved are the percolation model in equilibrium phase transitions,and DP,PCPD models in non-equilibrium phase transitions.For the equilibrium phase transitions,we choose a two-dimensional lattice as the learning object and the(1+1)dimensional case for the non-equilibrium phase transitions.The results show that if the order parameter of the phase transition model is the particle density,then the single potential variable of the autoencoder neural network is positively correlated with the order parameter.However,the percolation model produces a different result from the other models because its order parameter is a quantity related to the density of the number of clusters rather than the particle density.Therefore,the autoencoder neural network cannot extract the critical value of the percolation model.From above,we prove that the single latent variable of the autoencoder neural network does not necessarily represent the critical properties of the phase transition model unless its order parameter can be expressed in terms of particle density.Finally,we also apply a ML technique called transfer learning to study phase transitions.For our percolation and directed percolation models,transfer learning needs only a few of their configurations to be labeled.Using a training method named adversarial domain adaptation,we can obtain reasonable critical points and some critical exponents for both models.Adversarial domain adaptation has the advantage that both supervised and unsupervised learning do.Another significant result is that the adversarial domain adaptation method is also conditional in identifying phase transitions.To get better critical point information,we claim the input configuration should be associated with the order parameter.For the directed percolation model,the order parameter is particle density represented by its occupation sites.Therefore,we can obtain excellent prediction results by using transfer learning.However,the order parameter of percolation is a quantity related to the percolating cluster.It leads to a result that we must make feature engineering to the original configuration.We show that picking the configuration of maximum-cluster can heavily characterize the percolating transition through transfer learning.Using ML methods to study the reaction-diffusion processes of unitary and binary,we obtained many meaningful results in this thesis.In the case of small size,ML performs well compared to conventional MC simulations.As algorithms and computational power continue to be updated,ML will have better performance in statistical physics.