Deep Learning Methods of Contact Manifolds Reconstruction for Non-equilibrium Statistical Physics

Random walk with volume and surface reinforcement is a novel model of non-Markovian discrete random walk on a three-dimensional lattice. The idea of the model is that the next step of the walk depends on the entire previous trajectory: we differentiate between probabilities of (i) steps directed to already visited points of the lattice (volume reinforcement), (ii) steps along the surface of the previously visited volume (surface reinforcement), and (iii) all others steps. Such a behavior resembles volume interactions in polymer systems; however, in contrast to the problem of a polymer coil (globule), the system under consideration is essentially nonequilibrium and its dynamics cannot be described by Hamilton's equations. In a certain range of parameter values, the random walk trajectories demonstrate the presence of hierarchical folded structures that resemble the three-dimensional structure of DNA in chromosomes. Due to the non-Markovian nature of the walk, it is difficult to investigate it analytically or using traditional methods of finite-size scaling. We propose a technique based on ideas from contrastive learning, an active research area in computer vision. The deep learning model tries to distinguish between two given microstates and determine if they were generated with the same values of macroscopic parameters. It is assumed that such a distinction is relatively easy for trajectories belonging to different morphologic phases, and much more difficult for trajectories obtained for different values of parameters from the same phase. The application of this idea will make it possible to construct a morphological (phase) diagram of the system based on data on relatively short trajectories, which will require significantly less computational resources than the finite-size scaling method. The developed method was generalized and adapted for the problem of reconstructing the thermodynamic equation of state from an ensemble of microstates.