The relentless evolution of computing—from classical algorithms to artificial intelligence (AI), brain-inspired architectures, and quantum systems—demands a reimagining of the mathematical foundations that underpin these transformative technologies. Edited by Prof. Shi Jin from Shanghai Jiao Tong University, this volume bridges abstract mathematical theory and cutting-edge computational practice, equipping researchers, engineers, and students with fundamental understanding to navigate and shape the future of scientific computing.This book unifies interdisciplinary advances in quantum computation, neural and brain-inspired systems, AI-driven molecular modeling, and data-intensive particle dynamics under a cohesive mathematical framework. It synthesizes insights from 9 meticulously structured chapters, each authored by leading experts, to address the mathematical challenges and innovations arising in:- Quantum advantage for solving partial differential equations (PDEs),- Biologically plausible AI modeling brain dynamics and protein engineering,- Statistical rigor in randomized experiments and observational studies,- Operator learning for kinetic equations and interacting particle systems,- Graph/hypergraph neural networks via phase-transition physics.The book aims to reveal the mathematical "language" of next-generation computing paradigms. As computing paradigms fracture into specialized niches, this volume is a unifying compass. It transforms isolated breakthroughs—such as particle-based graph networks, operator learning for chemistry, or quantum PDE solvers—into a coherent mathematical arsenal. For mathematicians, it reveals uncharted problems in computation; for engineers, it provides rigor to harness AI, quantum, and bio-inspired tools; for students, it maps the emerging landscape where equations meet evolution.
