Unveiling convolutional neural networks in surrogate modeling (and more!)

Title

Unveiling convolutional neural networks in surrogate modeling (and more!)

Speaker

Nicola Rares Franco - Politecnico di Milano, MOX laboratory

Abstract

Surrogate models are suitable emulators that aim at approximating the behavior of complex systems at a reduced computational cost. They play a vital role in fields such as uncertainty quantification and optimal control, particularly in scenarios involving parametrized partial differential equations (PDEs), where they serve as a cheaper alternative to expensive numerical solvers. Recently, Deep Learning has become increasingly popular in this context, providing researchers with new powerfull data-driven approaches to surrogate modeling, from DeepONets and Fourier Neural Operators to models based on deep convolutional autoencoders. In this talk I will focus on the latter class of approaches, with a major emphasis on Convolutional Neural Networks (CNNs). In particular, by casting the problem in the framework of operator learning, I will present suitable error bounds that illustrate the role played by each hyperparameter in a convolutional architecture, ultimately unveiling how -and why- CNNs actually work. To this end, I will first present an analysis of the sole approximation error, ignoring model training and optimization; then, I will present some novel results that aim at bridging this gap, discussing practical issues, such as the choice of the training size and the loss function. Finally, I will conclude with a brief discussion about the broader picture, mentioning how surrogate models can be combined with numerical solvers to optimaly manage computational resources, and how prior knowledge can help us in constructing physically consistent architectures.

Bio

Nicola R. Franco is a Postdoc researcher at the Department of Mathematics of Politecnico di Milano (MOX Laboratory). He graduated at Università degli Studi di Milano (Faculty of Mathematics), and later obtained his Ph.D. in "Mathematical models and methods in engineering" at Politecnico di Milano (2023). His research focuses primarily on deep learning approaches to model order reduction, addressing both theoretical and practical aspects, with a major bias towards oncological applications, specifically personalized treatment planning of radiotherapy.

When

Thursday, April 11th, 14:30

Where

DIBRIS/DIMA, Via Dodecaneso 35, Room 704