دانلود کتاب روش‌های یادگیری عمیق فیزیک ریاضی – جلد اول مسائل مستقیم و معکوس

The rapid progress of Artificial Intelligence (AI) and Deep Learning (DL) is reshaping many scientific disciplines, including Mathematical Physics–a field fundamentally governed by differential equations. Traditionally, problems in Mathematical Physics have been addressed through analytical techniques and numerical solvers that integrate the governing equations. As systems become more complex, nonlinear, or chaotic, however, classical methods can struggle to capture full dynamics or to infer governing laws directly from data. Deep Learning complements these approaches by providing flexible, data-driven surrogates and physics-informed formulations that scale to high complexity while respecting the underlying differential structure. The year 2024 is particularly significant because it coincides with the awarding of the Nobel Prizes in Physics [90] and Chemistry [89] to research that has made pioneering use of Deep Learning methods to tackle problems that had resisted solution for decades. These major achievements show that Deep Learning is no longer just a computer tool. It has become a key part of modern science, helping researchers discover things that were once impossible to uncover. This book explores the emerging role of Deep Learning as a computational tool in Mathematical Physics. Neural architectures such as Neural Ordinary Differential Equations (Neural ODEs), Physics-Informed Neural Networks (PINNs), Hamiltonian Neural Networks (HNNs), and Lagrangian Neural Networks (LNNs) allow the direct encoding of physical principles into learning models. Deep Learning methods not only address direct problems–traditionally solved using analytical techniques or numerical simulations–but also open new avenues for solving inverse problems. They enable the inference of system parameters, conservation laws, and even the underlying governing equations directly from experimental or simulated data. Deep Learning offers unique advantages for classical mechanics problems. Architectures designed with physical insights can enforce conservation laws, handle complex boundary conditions, and generalize to unseen scenarios. This integration of Physics and machine learning not only improves predictive capabilities but also opens avenues for discovering new physical models directly from data. This book emphasizes both the practical implementation and the theoretical underpinnings of these methods. Through numerous examples and comparisons, the text illustrates how Deep Learning models compare to, complement or surpass traditional solvers in certain regimes. Many examples in this book include Keras implementations, either embedded in the text or linked to online Google Colab notebooks. These examples make strong class projects: readers can modify inputs and hyperparameters to observe how the outputs change. In some cases, the exposition is intentionally kept at a theoretical level, inviting the reader to implement the corresponding code. Keras is used throughout because it is widely adopted, well documented, and well supported within the TensorFlow ecosystem. Each chapter concludes with end-of-chapter exercises, several of which are suitable as course projects and reinforce the methods introduced in that chapter. This book is neither a primer on machine learning nor a physics textbook. It is intended for readers already comfortable with Deep Learning who want to see how modern architectures can be applied to problems in Mathematical Physics. The material is organized as a collection of worked examples—ranging from simple to challenging— that illustrate when Deep Learning helps, how to set it up, and what its practical limits are. The book targets senior undergraduates but is also suitable for graduate students, researchers, and practitioners in Physics, Applied Mathematics, Engineering, and Computer Science. The ideal reader is familiar with basic Classical Mechanics, differential equations, and introductory machine learning concepts. Prior experience with Python, TensorFlow, or Keras is helpful but not required; the book provides ample examples and guidance to support implementation. The book is organized into three main parts: The first part is the most elementary and it is an introduction to neural networks and its applications to mathematics problems. The second part focuses on using Deep Learning models to solve forward problems, including classical benchmarks such as the harmonic oscillator, planetary motion, and the simple pendulum. Performance is compared with classical numerical methods. The third part deals with inverse problems, exploring data-driven discovery of system parameters, physical laws, and conservation laws.