Reconstruction and Parameter Estimation of Dynamical Systems using Neural Networks

Dynamical systems can be loosely regarded as systems whose dynamics is entirely determined by en evolution function and an initial condition, being therefore completely deterministic and a priori predictable. Nevertheless, their phenomenology is surprisingly rich, including intriguing phenomena such as chaotic dynamics, fractal dimensions and entropy production. In Climate Science for example, the emergence of chaos forbids us to have meteorological forecasts going beyond fourteen days in the future in the current epoch and therefore building predictive systems that overcome this limitation, at least partially, are of the extreme importance since we live in fast-changing climate world, as proven by the recent not-so-extreme-anymore climate phenomena. At the same time, Machine Learning techniques have been widely applied to practically every field of human knowledge starting from approximately ten years ago, when essentially two factors contributed to the so-called rebirth of Deep Learning: the availability of larger datasets, putting us in the era of Big Data, and the improvement of computational power. However, the possibility to apply Neural Networks to chaotic systems have been widely debated, since these models are very data hungry and rely thus on the availability of large datasets, whereas often Climate data are rare and sparse. Moreover, chaotic dynamics should not rely much on past statistics, which these models are built on. In this thesis, we explore the possibility to study dynamical systems, seen as simple proxies of Climate models, by using Neural Networks, possibly adding prior knowledge on the underlying physical processes in the spirit of Physics Informed Neural Networks, aiming to the reconstruction of the Weather (short term dynamics) and Climate (long term dynamics) of these dynamical systems as well as the estimation of unknown parameters from Data.