Abstract: A dynamical physical system of particles represented by input data representing the phase space state of the particles over time is characterised by deriving a feature vector in respect of each particle comprising plural metrics that each describe a change in the phase space state of the particle over time. A classification of particles into plural classes is performed by applying a machine learning technique that operates on the feature vectors of the particles, and outputting classification data representing the classification.

Abstract: Machine learning is performed on input data representing a dynamical statistical system of entities having plural primary variables that vary time. A distribution function over time of the density of entities in a phase space, whose dimensions are the primary variables and secondary variables dependent on the rate of change of the primary variables, is derived and encoded as a sum of contour functions over time describing the contour in phase space of plural phaseons which are entities of a model of the dynamical statistical system that are localised in the phase space. Machine learning is performed on the encoded distribution function and/or at least one field in the effective configuration space whose dimensions are the primary variables, derived from the encoded distribution function. The encoding of the distribution function provides a representation which improves the performance of the machine learning techniques by simplifying hyperparameter optimisation.