Solving Differential Equations and Inverse Problems with Ai: With Problems and Solutions - Part I

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.