Deep Learn Method Mathe Phy (V1) eBook

This book explores how Artificial Intelligence and Deep Learning are transforming Mathematical Physics, offering modern data-driven tools where traditional analytical and numerical methods fall short. As physical systems grow more complex or chaotic, deep learning provides efficient surrogates and physics-informed models capable of capturing dynamics and uncovering governing laws directly from data.

This book introduces Neural ODEs, Physics-Informed Neural Networks (PINNs), and Hamiltonian and Lagrangian Neural Networks, showing how they enhance classical mechanics and PDE solvers for both forward and inverse problems. With Keras code examples, Google Colab notebooks, and practical exercises, this book serves researchers and students in physics, mathematics, and engineering seeking a concise, hands-on guide to applying deep learning in physical systems.

Contents:

• Introduction to Neural Networks and Applications:
  • Review of Feed-forward Neural Networks
  • Deep Learning for Solving Equations
• Direct Problems: Predicting Motion with Deep Learning:
  • Introduction to Classical Mechanics with Neural Nets
  • Lagrangian Systems
  • Hamiltonian Systems
  • Conservation Laws
  • Hamilton–Jacobi Theory
  • Free-boundary Value Problems
  • Classical PDEs of Mathematical Physics
• Inverse Problems: Inferring Physics from Data:
  • Introduction to Inverse Problems in Mechanics
  • Identifying Conservation Laws
  • Parameter Estimation
  • Equation Discovery
  • Inverse Problems for Classical PDEs
  • Special Structures
  • Further Applications
  • Miscellaneous Exercises
• Miscellaneous Exercises
• Bibliography
• Index

Readership: Advanced undergraduate and graduate students, researchers and practitioners in the fields of AI, Mathematical Physics, Computer Science, and Engineering.