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Neural networks are very powerful learning systems, but they do not readily generalize from one task to the other. This is partly due to the fact that they do not learn in a compositional way, that is, by discovering skills that are shared…
This article surveys the noncommutative-geometric (NCG) approach to fundamental physics, in which geometry is encoded spectrally by a generalized Dirac operator and where dynamics arise from the spectral action. I review historically how…
We show that all kinds of biasing of cosmological phase transitions produce qualitatively new type of domain wall networks. The biased networks consist of compact, finite size, bag-like wall structures and exhibit a generic instability. The…
It has been recently observed in much of the literature that neural networks exhibit a bottleneck rank property: for larger depths, the activation and weights of neural networks trained with gradient-based methods tend to be of…
This paper is a physicist's review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here we present a unified, graph-based view of all archetypical models of such universality (billiards,…
Let $\Omega \subset\widehat{\mathbb{C}}$ be a multiply connected domain, and let $\pi\colon \mathbb{D}\to\Omega$ be a universal covering map. In this paper, we analyze the boundary behaviour of $\pi$, describing the interplay between radial…
Supersymmetric U(Nc) gauge theory with Nf massive hypermultiplets in the fundamental representation admits various BPS solitons like domain walls and their webs. In the first part we show as a review of the previous paper hep-th/0412024…
Graph neural networks (GNNs) have achieved remarkable success in various domains, yet they often struggle with domain adaptation due to significant structural distribution shifts and insufficient exploration of transferable patterns. One of…
Novel imaging and neurostimulation techniques open doors for advancements in closed-loop control of activity in biological neural networks. This would allow for applications in the investigation of activity propagation, and for diagnosis…
Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within…
We explore different strategies to integrate prior domain knowledge into the design of a deep neural network (DNN). We focus on graph neural networks (GNN), with a use case of estimating the potential energy of chemical systems (molecules…
The magnetic structure in the Galactic disk, the Galactic center and the Galactic halo can be delineated more clearly than ever before. In the Galactic disk, the magnetic structure has been revealed by starlight polarization within 2 or 3…
The grain boundary (GB) energy is a quantity of fundamental importance for understanding several key properties of graphene. Here we present a comprehensive theoretical and numerical study of the entire space of symmetric and asymmetric…
We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of…
We study a classical integrable (Neumann) model describing the motion of a particle on the sphere, subject to harmonic forces. We tackle the problem in the infinite dimensional limit by introducing a soft version in which the spherical…
In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which…
We show that the real massive Klein-Gordon theory admits a description in terms of states on various timelike hypersurfaces and amplitudes associated to regions bounded by them. This realizes crucial elements of the general boundary…
Generalized splines are a simultaneous generalization of GKM theory -- which studies equivariant cohomology -- and classical splines, which provide piecewise approximations of functions. Generalized splines can also be understood via…
Graph neural networks (GNNs) have achieved remarkable success in processing graph-structured data across various applications. A critical aspect of real-world graphs is their dynamic nature, where new nodes are continually added and…
Magnetar phenomena are likely intertwined with the location and structure of magnetospheric currents. General-relativistic effects are important in shaping the force-free equilibria describing static configurations, though most studies have…