Statistical mechanics and learning problems in neural networks

My PhD thesis is based on Statistical Mechanics themes and their applications. In the second chapter I test the inverse problem method for a class of monomer-dimer statistical mechanics models that contain also an attractive potential and display a mean-field critical point at a boundary of a coexistence line. I obtain the inversion by analytically identifying the parameters in terms of the correlation functions and via the maximum-likelihood method. The precision is tested in the whole phase space and, when close to the coexistence line, the algorithm is used together with a clustering method to take care of the underlying possible ambiguity of the inversion. In the third chapter I perform some analysis in order to characterize statistical properties of the observed mobility of drosophilas expressing different kinds of proteins. In the fourth chapter I give an overview of the already existing algorithm Replicated Belief Propagation (RBP) deeply analyzing the equations which define the model. In the fifth chapter I apply the RBP in order to predict the congestion formation in the framework of complex systems physics. Traffic is a complex system where vehicle interactions and finite volume effects produce different collective regimes and phase transition phenomena. Such prediction can be a difficult problem due to the heterogenous behavior of drivers when the vehicle density increases. We propose a novel pipeline to classify traffic slowdowns by analyzing the features extracted from the fundamental diagram of traffic. I train the RBP and we provide a forewarning time of prediction related to the training set size. Then I compare my results with those of the most common classifiers used in machine learning analysis.