Anomalous diffusion characterization using machine learning methods

Resumen

During the last decades the use of machine learning and artificial intelligence have showed an exponential growth in many areas of science. The fact that computer's hardware has increased its performance while lowering the price and the availability of open source frameworks have enabled the access to artificial intelligence to a broad range of researchers, hence democratizing the access to artificial intelligence methods to the research community. It is our belief that multi-disciplinarity is the key to new achievements, with teams composed of researchers with different backgrounds and fields of specialization. With this aim, we focused this thesis in using machine learning, artificial intelligence, deep learing, all of them being understood as part of a whole concept we concrete in artificial intelligence, to try to shed light to some problems from the fields of mathematics and physics. A deep learning architecture was developed and successfully benchmarked with the characterization of anomalous diffusion processes. Whereas traditional statistical methods had previously been used with this aim, deep learing methods, mainly based on recurrent neural networks have proved to outperform these clasical methods. Our architecture showed it can precisely infer the anomalous diffusion exponent and accurately classify trajectories among a given set of underlaying diffusion models. While recurrent neural networks irrupted in the recent years, convolutional network based models had been extensively tested in the field of image processing for more than 15 years. There exist many models and architectures, pre-trained and set to be used by the community. No further investigation needs to be done since the architecture have proved their value for years and are very well documented in the literature. Our goal was being able to used this well-known and reliable models with anomalous diffusion trajectories. We only needed to be able to convert a time series into an image, which we successfully did by applying gramian angular fields to the trajectories, focusing on short ones. To our knowledge this is the first time these techniques were used in this field. We show how this approach outperforms any other proposal in the underlaying diffusion model classification for short trajectories. Besides physics it is maths. We used our recurrent neural networks architecture to infer the parameters that define the Wu Baleanu trajectories. We show that our proposal can precisely infer both the mu and nu parameters with a reasonable confidence. Being the first time, to the best of our knowledge, that such techniques were applied to this scenario. We extend this work to the discrete delayed fractional equations, obtaining similar results in terms of precision. Additionally, we showed that the same architecture can be used to discriminate delayed from non-delayed trajectories with a high confidence. Finally, we also searched fractional discrete models. We have considered Lubich's quadrature time-stepping schemes instead of the classical Euler scheme of order 1. As the first study with this new paradigm, we compare the bifurcation diagrams for the logistic and sine maps obtained from Euler discretizations of orders 1, 2, and 1/2.