Computational Differential Equations with AI

This book provides cutting-edge machine learning (ML) methods, including Physics-Informed Neural Networks (PINNs), Neural Operators, and other ML methods, for solving ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic systems. Differential equations (DEs) are the basic foundation for modeling real-world problems in various fields such as physics, engineering, finance, and biology. Solving DEs requires complicated mathematical methods; however, ML is now a viable, innovative, and alternative technique. This book aims to bridge the gap between DEs and ML by explaining how to utilize neural networks, physics-informed models, and other artificial intelligence (AI) based techniques to solve DEs more efficiently and accurately. With the use of ML techniques, readers can also uncover hidden patterns within the data of the problem. The authors utilize Python throughout to implement and demonstrate the methods behind the various presented examples. This book is an ideal choice for academic researchers, engineers, data scientists, and others who are interested in ML methods and real-world applications to democratize next-generational computational mathematics. In addition, this book: Unifies PINNs, neural operators, and hybrid methods and discusses trade-offs (accuracy vs. computational cost) Blends classical DE solvers with pure ML solvers and includes ML-based numerical methods to solve problems Discusses how AI is transforming methods to solve differential equations