Shell models

Most of my publications deal with shell models of turbulence and related models.

A shell model is an idealization of Navier-Stokes or Euler equations. It is an infinite systems of ODEs usually indexed by the positive integers or by a regular tree. The equations are nearest-neighbour coupled, have quadratic nonlinearities and coefficients growing exponentially fast.

The tipical derivation is as follows. Consider the velocity field of some solutions of Navier-Stokes equations, write it in Fourier components and then collect in a single component indexed by n the energy of all the Fourier modes with wavenumber k in a spherical shell of radius between 2n-1 and 2n.

The equations for these shell components are not realistic, since the non-linear interaction beetween components can be arbitrarily chosen in several different ways, nevertheless they capture some of the statistical properties and features of three-dimensional turbulence, since they are usually taylored to model the cascade of energy to higher and higher wavenumbers.

The introduction of Stochastic inviscid shell models: well-posedness and anomalous dissipation is a good place to start.

Research on shell models.

Machine Learning

I have always had small projects and activities in multivariate statistics and machine learning, and recently I started to do original research on the field with two large programs: SAI and COTAN.

SAI (Sensible Artificial Intelligence) is a wide project on Deep Reinforcement Learning that aims at improving AlphaGo Zero, by generalizing the winrate estimation to a parametric family of sigmoid curves.

A side project is to understand and improve various aspects of general feed forward ANNs.

SAI project.

COTAN (CO-expression Tables ANalysis) is a multidisciplinary project in big data from biology, in particular it is devoted to the study of single-cell trascriptomics datasets. It is based on a new dicrete probability model and its derived statistics inference framework. It shows SOTA performace on determining the genes co-expression and in clustering genes by function.

COTAN project.


Older research directions included statistical properties of random error correcting codes and branching process representations of non-linear ODEs.

Other papers.

Vangogh Starry Night Alpha Zero fan art