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 eective 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.

Covers a broad spectrum of hot topics Authored by some of the top experts in their fields Contains a chapter on scientific machine learning for differential problems

Autorentext

Giuseppe Coclite is a full professor at the Politecnico di Bari in Italy. His research interests include nonlinear PDEs and control theory. 

Enrique Zuazua is a full professor at the Friedrich-Alexander-Universität Erlangen-Nürnberg in Germany. His research interests include nonlinear PDEs and control theory. 

Alberto Bressan is a full professor at the Penn State University in the United States. His research interests include nonlinear PDEs and control theory. 

**Leon Bungert **is an associate professor at the University of Würzburg in Germany. His research interests include machine learning and applied analysis.

Alfio Quarteroni is an emeritus professor at the Politecnico di Milano in Italy and EPFL in Switzerland. His research interests include numerical analysis and deep learning. 

Guido De Philippis is a full professor at the Universita di Padova in Italy. His research interests include the calculus of variations, transport theory and geometric measure theory. 

Klappentext

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 e ective 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. 

Inhalt

Chapter 1. Introduction.- Chapter 2. Control Problems for Propagation Fronts and Moving Sets.- Chapter 3. PDEs in Data Science: From Graph Learning to Adversarial Training.- Chapter 4. The Bernoulli Free Boundary Problem.- Chapter 5. Scientific Machine Learning for differential problems: Integrating physics-based and data-driven approaches.