Research statement

I am currently doing research in two fields: 

• The study of the controllability and observability properties of Partial Differential Equations, and on the cost of the optimal controls. 

• The mathematical understanding of Neural Networks focusing on their application to the numerical resolution of Partial Differential Equations and to Approximation Theory.

I will now explain what these research fields consist of, and my contribution to them:

• The control of systems governed by Partial Differential Equations is a modern research area of applied mathematics, leading to rich, innovative research and to powerful ideas and methods. My work on the parabolic and dispersive equations focuses on the proof of the existence (or of the non-existence) of controls and on obtaining estimates for the cost of such control. A general mathematical formulation of the question I am interested in is the following: given a PDE system, I want to find an input (a force or a boundary control in a given space) that allows to control one aspect of the system (the final state, a trace, the average when there is a parameter for which only its probability distribution is known, etc.) or, alternatively, to give a proof of the non-existence of such input. When the answer is positive, I will try to estimate the amount of energy that such a force requires. I have done research in this topic since I started my PhD thesis. 

I have worked with a great variety of PDEs that model many physical phenomena. In particular: the heat equation, which governs heat diffusion, as well as other processes, such as particle diffusion or propagation of action potential in nerve cells; the penalized Navier-Stokes system, a PDE which behaves well numerically and approximates the incompressible Navier-Stokes system, which models the motion of Newtonian viscous fluids; the KdV equation, which describes waves on shallow water surfaces; and the Stefan system, which models the evolution of the boundary between two phases of material undergoing a phase change (for example, the melting of ice).

To know more about my contributions to this topic, you may watch any of these talks:

• • • My talk about parabolic equations with vanishing for the Bilbao Analysis and PDE group.

    • My talk about average controllability of the heat equation for Control in Times of Crisis.  

    • My talk explaining results about the controllability of the heat equation for the Chair of Dynamics, Control, and Numerics of the Friedrich-Alexander-Universität Erlangen-Nürnberg.

Moreover, you can also consult the following blog-post summarizing some of my papers:

• • • The following post is about the average controllability of the heat equation. 

    • The following post is about the controllability of the heat equation with vanishing diffusion in networks. 

• Artificial neural networks are a computational model that mimic the functioning of the human brain. They consist of a set of units, called artificial neurons, connected together to transmit signals to each other. From a mathematical point of view, neural networks are a family of functions with multiple applications. In the field of applied mathematics, networks play a fundamental role in allowing the resolution of complex mathematical problems efficiently and accurately. Networks are used to solve optimization problems, inverse problems, function approximation and control theory, as well as for data analysis and prediction of future results. Neural networks are expected to continue to evolve and improve their performance in this field.

The problem on neural networks that currently interests me the most is the estimation of the size that these networks must have to approximate a certain function or the solutions of Partial Differential Equations. I am also interested in obtaining controllability properties of Differential Equations whose dynamics are described by neural networks. 

So far, I have a paper on Neural Networks, and I have directed several Bachelor Thesis and a Master Thesis on Neural Networks. Moreover, I am currently supervising a PhD Thesis on Neural Networks. 

Finally, I would like to add that my potential research interests include mathematical understanding of the Partial Differential Equations and of the properties of functions through Mathematical Analysis.

For further details, my list of publications can be consulted. 

Research statement updated in the 21st of October, 2024