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${\mathrm{Nb}}_3\mathrm{Sn}$film coatings have the potential to drastically improve the accelerating performance of Nb superconducting radiofrequency (SRF) cavities in next-generation linear particle accelerators. Unfortunately, persistent ${\mathrm{Nb}}_3\mathrm{Sn}$stoichiometric material defects formed during fabrication limit the cryogenic operating temperature and accelerating gradient by nucleating magnetic vortices that lead to premature cavity quenching. The SRF community currently lacks a predictive model that can explain the impact of chemical and morphological properties of ${\mathrm{Nb}}_3\mathrm{Sn}$defects on vortex nucleation and maximum accelerating gradients. Both experimental and theoretical studies of the material and superconducting properties of the first 100 nm of ${\mathrm{Nb}}_3\mathrm{Sn}$surfaces are complicated by significant variations in the volume distribution and topography of stoichiometric defects. This work contains a coordinated experimental study with supporting simulations to identify how the observed chemical composition and morphology of certain Sn-rich and Sn-deficient surface defects can impact the SRF performance. ${\mathrm{Nb}}_3\mathrm{Sn}$films were prepared with varying degrees of stoichiometric defects, and the film surface morphologies were characterized. Both Sn-rich and Sn-deficient regions were identified in these samples. For Sn-rich defects, we focus on elemental Sn islands that are partially embedded into the ${\mathrm{Nb}}_3\mathrm{Sn}$film. Using finite element simulations of the time-dependent Ginzburg-Landau equations, we estimate vortex nucleation field thresholds at Sn islands of varying size, geometry, and embedment. We find that these islands can lead to significant SRF performance degradation that could not have been predicted from the ensemble stoichiometry alone. For Sn-deficient ${\mathrm{Nb}}_3\mathrm{Sn}$surfaces, we experimentally identify a periodic nanoscale surface corrugation that likely forms because of extensive Sn loss from the surface. Simulation results show that the surface corrugations contribute to the already substantial drop in the vortex nucleation field of Sn-deficient ${\mathrm{Nb}}_3\mathrm{Sn}$surfaces. This work provides a systematic approach for future studies to further detail the relationship between experimental ${\mathrm{Nb}}_3\mathrm{Sn}$growth conditions, stoichiometric defects, geometry, and vortex nucleation. These findings have technical implications that will help guide improvements to ${\mathrm{Nb}}_3\mathrm{Sn}$fabrication procedures. Our outlined experiment-informed theoretical methods can assist future studies in making additional key insights about ${\mathrm{Nb}}_3\mathrm{Sn}$stoichiometric defects that will help build the next generation of SRF cavities and support related superconducting materials development efforts. Published by the American Physical Society2024 

We develop information-geometric techniques to analyze the trajectories of the predictions of deep networks during training. By examining the underlying high-dimensional probabilistic models, we reveal that the training process explores an effectively low-dimensional manifold. Networks with a wide range of architectures, sizes, trained using different optimization methods, regularization techniques, data augmentation techniques, and weight initializations lie on the same manifold in the prediction space. We study the details of this manifold to find that networks with different architectures follow distinguishable trajectories, but other factors have a minimal influence; larger networks train along a similar manifold as that of smaller networks, just faster; and networks initialized at very different parts of the prediction space converge to the solution along a similar manifold. 

We develop information geometric techniques to understand the representations learned by deep networks when they are trained on different tasks using supervised, meta-, semi-supervised and con- trastive learning. We shed light on the following phenomena that relate to the structure of the space of tasks: (1) the manifold of probabilistic models trained on different tasks using different represen- tation learning methods is effectively low-dimen- sional; (2) supervised learning on one task results in a surprising amount of progress even on seem- ingly dissimilar tasks; progress on other tasks is larger if the training task has diverse classes; (3) the structure of the space of tasks indicated by our analysis is consistent with parts of the Word- net phylogenetic tree; (4) episodic meta-learning algorithms and supervised learning traverse differ- ent trajectories during training but they fit similar models eventually; (5) contrastive and semi-su- pervised learning methods traverse trajectories similar to those of supervised learning. We use classification tasks constructed from the CIFAR- 10 and Imagenet datasets to study these phenom- ena. Code is available at https://github.com/grasp- lyrl/picture of space of tasks. 

Abstract Complex models in physics, biology, economics, and engineering are often sloppy , meaning that the model parameters are not well determined by the model predictions for collective behavior. Many parameter combinations can vary over decades without significant changes in the predictions. This review uses information geometry to explore sloppiness and its deep relation to emergent theories. We introduce the model manifold of predictions, whose coordinates are the model parameters. Its hyperribbon structure explains why only a few parameter combinations matter for the behavior. We review recent rigorous results that connect the hierarchy of hyperribbon widths to approximation theory, and to the smoothness of model predictions under changes of the control variables. We discuss recent geodesic methods to find simpler models on nearby boundaries of the model manifold—emergent theories with fewer parameters that explain the behavior equally well. We discuss a Bayesian prior which optimizes the mutual information between model parameters and experimental data, naturally favoring points on the emergent boundary theories and thus simpler models. We introduce a ‘projected maximum likelihood’ prior that efficiently approximates this optimal prior, and contrast both to the poor behavior of the traditional Jeffreys prior. We discuss the way the renormalization group coarse-graining in statistical mechanics introduces a flow of the model manifold, and connect stiff and sloppy directions along the model manifold with relevant and irrelevant eigendirections of the renormalization group. Finally, we discuss recently developed ‘intensive’ embedding methods, allowing one to visualize the predictions of arbitrary probabilistic models as low-dimensional projections of an isometric embedding, and illustrate our method by generating the model manifold of the Ising model.