Giuseppe Maria Coclite – författare

This volume includes lectures presented at the CIME School on “PDEs, Control and Deep Learning”, held in Cetraro (Italy) from July 22 to 26, 2024. It provides a comprehensive and up-to-date view of the diverse and rapidly evolving field of nonlinear partial differential equations (PDEs), with an emphasis on modeling, analysis, control, and deep learning aspects. The theory of PDEs interacts closely with almost all areas of physics and many branches of mathematics. As explicit solutions of PDEs are rarely available (except in the simplest cases), numerical approximations play a central role in their study. Machine learning, particularly through artificial neural networks, introduces powerful methods for function approximation through layered structures of interconnected units (neurons) that combine linear transformations and nonlinear activations. Deep learning (the use of neural networks with many hidden layers) has proven to be remarkably effective in a wide range of applications. At the same time, recent advances in PDEs and control theory are beginning to inform machine learning, providing new theoretical perspectives.The book will be a valuable resource for PhD students and researchers seeking to deepen their understanding of partial differential equations, control, and their connections to modern machine learning.

This book are notes prepared for the PhD courses that the author has been teaching during the last 10 years. The material available in the already existing literature (papers and essays) has been collected in this unique text, presenting the results with all the details for the reader’s convenience, fixing a unified notation, and providing a consistent framework for the subject. These notes cover many of the arguments that usually can be found in high level essays, where the proofs are simply sketched, and in papers, which are not easily available and not always self-contained.This book is intended for1. PhD students in Mathematics, Physics and Mechanical Engineering in order to learn the basic features of nonlinear scalar equations,2. researchers interested in nonlinear hyperbolic PDEs in order to learn the details behind some known and deep results on nonlinear scalar equations,3. teachers of courses on nonlinear PDEs.The readers are expected to know the basic measure theory and Sobolev spaces.