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.
Introduction to Differential Equations (DEs) and Machine Learning.-
Mathematical Preliminaries.- Machine Learning Basics for DEs.-
Physics-Informed Neural Networks (PINNs).- Neural Operators and DeepONets.-
Extreme Learning Machines and Random Feature Methods.- Neural Ordinary
Differential Equations (Neural ODEs).- Reinforcement Learning for Dynamic
Systems.- Solving Stochastic Differential Equations (SDEs) with Machine
Learning.- Appendices: A: Software Tools (PyTorch, TensorFlow, JAX);
B: Datasets and Benchmark Problems; C: Mathematical Reference (Key Theorems,
Notation).
Snehashish Chakraverty, Ph.D., is a senior Professor (HAG) in the Department of Mathematics at the National Institute of Technology Rourkela, India. He is a globally recognized academician and researcher with over 30 years of experience in the fields of mathematical modeling, structural mechanics, uncertainty quantification, and computational methods. Dr. Chakraverty has authored more than 35 books and 340 peer-reviewed journal articles published by reputed publishers. He has also served as the editor and editorial board member of numerous international journals and has received numerous national and international awards for his contributions to mathematical modeling and applied mechanics. His recent research spans soft computing, AI techniques in mechanics, and computational modeling of smart structures.
Sandeep Kumar Samota, Ph.D., is a Senior Research Fellow in the Department of Mathematics at the National Institute of Technology Rourkela, India. His research focuses on applying machine learning methods to address problems in dynamic systems, utilizing artificial neural networks, conducting data analysis, and predicting poverty through machine learning techniques.
Reema Gupta, Ph.D., is a Senior Research Fellow in the Department of Mathematics at the National Institute of Technology Rourkela, India. Her research focuses on solving various types of stochastic models using a range of numerical techniques, including the operational matrix method, the Galerkin method, and the collocation method.
