This open access book offers a bold and unifying vision of how scientific knowledge can be expressed through universal mathematical structures. Moving from classical dimensional analysis to modern probabilistic and data-driven frameworks, this book reveals deep connections between seemingly disparate fields.

Beginning with the foundations of the Buckingham Pi theorem, the authors guide the reader toward a broader perspective where functional equations, copulas, Bayesian networks, and functional networks emerge as parts of a single coherent framework. The work introduces powerful tools for modeling complex systems, emphasizing how relationships between variables can be captured, generalized, and extended across disciplines.

Bridging theory and application, the book explores engineering, physics, and data science problems while introducing innovative concepts such as opacity technique to reveal conditional-density structures in data clouds, conditional modeling with events of positive probability, and new approaches to artificial intelligence grounded in structure rather than blackbox methods.

Designed for researchers, engineers, and advanced students, this book is both a reference and a source of new ideas. It challenges conventional modeling approaches and invites readers to rethink how scientific laws are discovered, represented, and applied in an increasingly data-rich world.

Enrique Castillo (1946)

Civil Engineer, MSc Mathematics, PhD Civil Eng. from Northwestern and Polytechnic University of Madrid. Supernumerary Member of the Royal Spanish Academy of Engineering and Royal Spanish Academy of Sciences. Spanish National Awards in Engineering (2010) and Statistics (2022). Dr Hc from the University of Oviedo, University of Castilla La Mancha and University of Valparaiso. author of a number of books on different fields of Statistics (Bayesian theory, Extreme Value Statistics, Functional networks), Functional equations, Safety in Construction, Traffic and Railways modeling.

Alfonso Fernández Canteli (1945)

Mechanical Engineer, Postgrade studies and assistant at the ETH Zürich, PhD Mech. Eng. Engineering from Polytechnic University of Madrid . Professor Emeritus at the University of Oviedo, Spain. RAI Laureate Engineer award and ESIS Fellow in 2022. His research focuses on probabilistic modeling of fracture and fatigue and damage accumulation design.