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This book offers a comprehensive exploration of the theoretical foundations of deep learning, bridging the gap between its ever increasing empirical success and rigorous mathematical understanding. With a focus on interdisciplinary approaches, it illuminates the interplay between mathematics, statistics, and computer science to address pressing challenges in deep learning research. While deep learning continues to revolutionize industries such as healthcare, gaming, and autonomous systems, its applications in solving mathematical problems like inverse problems and partial differential equations mark a paradigm shift in scientific inquiry. Yet, this rapid progress often outpaces our theoretical understanding, leaving critical questions about expressivity, optimization, robustness, and fairness unanswered.

This book aims to close this gap by presenting perspectives from mathematics, statistics, and applications. This interdisciplinary endeavor draws upon diverse mathematical frameworks, including algebraic geometry, differential geometry, stochastics, functional analysis, and topology, alongside fundamental tools from statistics and theoretical computer science. By synthesizing these fields, the book not only advances the theoretical foundation of deep learning but also sets the stage for novel applications across scientific domains. Targeted at researchers, advanced students, and professionals in mathematics, computer science, and related fields, this book serves as both an introduction to emerging theoretical insights and a roadmap for interdisciplinary collaboration. It is will be of value to anyone seeking to understand or to contribute to the future of deep learning.