The analysis of complex systems at nano- and micro-scales often requires their numerical simulation. Atomistic simulations, that rely on solving Newton's equation for each component of the system, despite being exact, are often too computationally expensive. In this work, firstly we analyse the properties of confined systems by extracting mesoscopic information directly from particles coordinate.

Then, taking advantage of Mori-Zwanzig projector operator techniques and advanced data-analysis tools, we present a novel approach to parametrize non-Markovian coarse-graining models of molecular system.
We focus on the parametrization of the memory terms in the stochastic Generalized Langevin Equation through a deep-learning approach.

Moreover, in the framework of Dynamical Density Functional Theory (DDFT) we derive a continuum non-Markovian formulation, able to describe, given the proper free-energy, the physical properties of an atomistic system.
Comparisons between molecular dynamics, fluctuating dynamical density functional theory and fluctuating hydrodynamics simulations validate our approach.

Finally, we propose some numerical schemes for the simulation of DDFT with additional complexities, i.e. with stochastic terms and non-homogeneous non-constant diffusion.