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.