Engineering solutions have evolved from Wiener’s closed-form analytical formulas and Kalman-Bucy’s dy-namic state-space algorithms to explicit numerical optimization and modern data-driven neural models. However, purely empirical black-box models discard decades of domain insights, demanding prohibitive amounts of annotated data and compute to relearn fundamental principles from scratch. Along a continuum between engineering-based white-box and empirical black-box models, we explore a gray-box framework and two versions along this continuum. In the first, we embed prior knowledge into the network architecture, so that intrinsic properties of the solution are guaranteed by design. For example, when the solution is known to be the gradient of a convex function, we introduce gradient networks (GradNets), whose architecture guarantees that the (trained) learned map is the gradient of a con-vex function, even though the underlying convex function is never known, showing that it overcomes several short-comings of existing approaches and can lead to substantial gains. In the second approach, at the extreme of the continuum, we demonstrate that, through careful architectural design, deep neural networks can learn directly from basic principles in a completely unsupervised and data-free setting—requiring zero training, labeled, or annotated data. By embedding system dynamics directly into the network topology, the architecture naturally converges to clas-sical analytical solutions. We illustrate this by showing how an untrained deep architecture autonomously recovers the optimal Kalman gain of the Kalman-Bucy filter directly from first principles.
José M. F. Moura, Philip L. and Marsha Dowd University Professor at Carnegie Mellon University (CMU), holds degrees from IST (Portugal) and MIT. He has research interests in signal processing and learning. Two of his patents, co-invented with Aleksandar Kavčić, cover a method for bit detection in hard disk drives. They led to a $750 million settlement in 2016 between CMU and a chip manufacturer, and to a 2026 settlement with a second chip maker. Together, the two settlements cover nearly all hard disk drives sold worldwide since 2003, the drives found in practically every computer sold over that period, some 5.5 to 6 billion of them. He co-invented Graph Signal Processing, a research area pursued by hundreds of research groups across the world. His intelligent compiler SPIRAL was licensed by Intel, moving it from hand-coded to machine-generated scientific libraries. At CMU, he directs a major educational and research collaboration program with several universities in Portugal. He was 2019 IEEE President and CEO, his leadership being considered transformational, particularly, in the areas of financial and governance transparency, open access, and diversity and inclusion. He is a member of the Academy of Sciences, Portugal, and of the US National Academy of Engineering, and a Fellow from the US National Academy of Inventors, holding doctor honoris causa degrees from the University of Strathclyde (UK), the Universidade de Lisboa (Portugal), and York University (Canada). He received the IEEE Jack S. Kilby Signal Processing Medal and the IEEE Haraden-Pratt Award