Solving Differential Equations And Inverse Problems With Ai: With Problems And Solutions - Part I - Hardcover

Protopapas, Pavlos; Jimenez, Raul

Hardcover

ISBN 10: 1800619073 ISBN 13: 9781800619074

Publisher: WSPC (EUROPE), 2026

Synopsis

This book explores the exciting intersection of machine learning and differential equations (DEs), presenting modern techniques to solve one of the most fundamental mathematical challenges. DEs govern the laws of nature, appearing in contexts as diverse as Einstein's general relativity, human behavior, and financial markets. Despite their ubiquity, no general analytical method exists to solve them, making numerical computation the only viable approach. Over the past decade, advances in neural networks have opened a new approach: Physics-Informed Neural Networks (PINNs). These models transform DEs into trainable neural architectures, enabling solutions with remarkable flexibility and efficiency. Drawing on over ten years of lectures at Harvard University, the authors provide a comprehensive introduction to PINNs, covering the theoretical foundations, algorithmic constructions, and practical techniques needed to implement them. Readers will gain a thorough understanding of differential equations, numerical methods, neural network architectures, boundary and initial value problems, optimization and sampling methods, and transfer learning strategies. Whether you are a student, researcher, or practitioner, this book equips you with the knowledge and tools to explore and contribute to this rapidly growing field at the cutting edge of science and technology.

About this title

About the Authors

Raúl Jiménez (Madrid, 1967) earned his PhD at the Niels Bohr Institute, University of Copenhagen, in 1995, then held a PPARC Advanced Fellowship at the Royal Observatory, Edinburgh. He later joined the Physics & Astronomy departments at Rutgers University and the University of Pennsylvania. In 2008, he became a Professor at the ICC, University of Barcelona, as part of ICREA and was a senior research fellow at Princeton (2007–2009) and a Radcliffe Fellow at Harvard (2015–2016). A theoretical physicist, his research spans cosmology and astrophysics, from early-universe physics to stellar phenomena, aiming to connect theory with observations and understand fundamental laws of nature. His work is supported by the Simons Foundation, the ERC, and the Spanish Government.

Pavlos Protopapas is an educator and researcher in data science and astronomy. At Harvard, he teaches courses including CS109A, CS109B, Introduction to Data Science, Advanced Topics in Data Science, and MLOps, with past experience in Capstone courses, Deep Reinforcement Learning, and upcoming plans for Physics-Informed Neural Networks. His courses are also offered through the Harvard Extension School and HarvardX, and he has contributed to Data Science schools in Chile, Colombia, India, and Rwanda.

His research lies at the intersection of astronomy, machine learning, and statistics, focusing on data-driven and physics-informed models for complex astrophysical phenomena. He has applied deep learning to astronomical data, from time-series light curves to interferometric imaging, developing transformer-based architectures such as Spectromer and Astromer, and machine learning pipelines for the Event Horizon Telescope to infer black hole parameters. He also advances Physics-Informed Neural Networks (PINNs) and hybrid modeling frameworks for fluid dynamics and general relativity, with broader interests in model interpretability, uncertainty quantification, and scalable AI systems for scientific discovery.

Edition details

• Publisher: WSPC (EUROPE)
• Publication date: 2026
• Language: English
• ISBN 10: 1800619073
• ISBN 13: 9781800619074
• Binding: Hardcover
• Number of pages: 188