From operator learning to reduced order modeling and artificial intelligence: a mathematical journey

The idea of learning from physical simulations has now emerged as a promising avenue for addressing the computational challenges posed by many-query applications, such as optimal control, uncertainty quantification and more. Within the literature, we can identify at least three fundamental constituents: (i) operator learning, which leverages approximation theory and functional analysis to come up with rigorous constructions; (ii) reduced order modeling, which combines tools from numerical analysis and PDE theory; and (iii) artificial intelligence, with a stronger data-centric and statistical core.

In this talk, I will focus on the interplay between these three research areas, presenting a series of mathematical results that aim to reconcile domain heuristics with rigorous theoretical guarantees, in an attempt to provide a clearer picture of an otherwise highly complex panorama. To this end, I will first discuss the importance of (nonlinear) low-rank structures, presenting some numerical experiments and related theoretical results featuring deep autoencoders and adaptive basis methods. These will include considerations on the choice of the latent dimension, design of the architectures and a recent note on parametric regularity. As a second step, I will shift the focus to other relevant aspects such as model expressivity, sample complexity, and physical consistency. There, I will present some theoretical contributions and outline specialized strategies for the design of physically-compliant deep learning models. Lastly, I will provide a brief overview about ongoing activities, mostly focusing on the development of probabilistic surrogates via generative AI.

Contatto:
andrea1.manzoni@polimi.it