PINN with Python: An Introduction

Physics-Informed Neural Networks (PINNs) transforming how we solve complex scientific and engineering problems. This book is your essential guide to understanding this powerful technique, which elegantly combines the flexibility of neural networks with the fundamental rigor of physical laws.   PINNs embed partial differential equations (PDEs) and their associated boundary and initial conditions directly into a neural network's training process via a custom loss function. This means the neural network itself learns to obey the laws of physics! The solution function becomes a neural network model—a smooth, differentiable function built from affine transformations and nonlinear activation functions, allowing it to intrinsically satisfy strong-form PDEs. Crucially, the derivatives needed to enforce these physical constraints are computed with "autograd," a lightning-fast, machine-precision method inherent to modern machine learning libraries.   This comprehensive and practical book delivers: Core Theory & Formulations:Understand the foundational principles that make PINNs work. Essential Techniques:Learn the practical methods for building, training, and applying PINN models. Step-by-Step PyTorch Implementations:Get hands-on with Python code to implement PINNs from scratch. Collocation vs. Energy-Based PINNs:Discover the nuances between these two primary approaches, their strengths, and when to use each.   The book dives deep into applications across key PDE types critical to science and engineering: Static Problems:Tackle Poisson equations. Time-Dependent Systems:Solve Heat equations (parabolic) and Wave equations (hyperbolic). Eigenvalue Challenges:Solve Helmholtz equations. Beyond theoretical concepts, you'll explore in-depth case studies demonstrating how to construct effective PINN models for various PDE types and geometries. We also confront real-world hurdles, including how to construct the loss function, the normalization of equation systems, resolving convergence issues, and developing robust training strategies. Benefit directly from the author's research experiences, insights, and practical utility codes, all integrated into this invaluable resource.   Whether you're a researcher pushing boundaries, a student eager to grasp cutting-edge computational methods, or a practitioner seeking advanced solutions, this book will equip you with the basic tools and understanding to deploy PINNs effectively across a vast array of PDE-driven challenges.