Veranstaltungen

Deep neural networks (DNNs) are singular statistical models whose loss landscapes exhibit complex degeneracies - features that defy explanation through classical regular statistical theory. Drawing on tools from singular learning theory, we introduce the Local Learning Coefficient (LLC), a quantity that rigorously quantifies the degree of degeneracy in DNNs. While the LLC is rooted in the singular framework, it recovers familiar notions of model complexity in regular or "minimally singular" regimes. We develop a scalable estimator for the LLC and apply it across a range of architectures, including deep linear networks with up to 100M parameters, ResNet image classifiers, and transformers. Empirical results demonstrate that the LLC sheds light on a range of deep learning phenomena, including in-context learning in transformers and the competition between energy and entropy during training dynamics, as well as how standard training heuristics influence effective model complexity. Ultimately, the LLC provides a practical instantiation of singular learning theory in modern deep learning, offering new perspectives on the interplay between overparameterization, generalization, and parsimony.

Models of non-Newtonian fluids play an important role in science and engineering and their mathematical analysis and numerical approximation have been active fields of research over the past decade. This lecture is concerned with the analysis of numerical methods for the approximate solution of a system of nonlinear partial differential equations that arise in models of chemically-reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra-filtrate of blood plasma that contains hyaluronic acid, whose function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where the power-law exponent depends on a spatially varying nonnegative concentration function, expressing the concentration of hyaluronic acid, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations one has to derive a uniform Hölder norm bound on the sequence of approximations to the concentration in a setting where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely a bounded function with no additional regularity. This necessitates the development of a discrete counterpart of the De Giorgi--Nash--Moser theory, which is then used, in conjunction with various compactness techniques, to prove the convergence of the sequence of numerical approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration