相关论文: Polynomial diagrams for microstructure modelling

Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and…

In machine learning, graph embedding algorithms seek low-dimensional representations of the input network data, thereby allowing for downstream tasks on compressed encodings. Recently, within the framework of network renormalization,…

Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph…

The theoretical models providing mathematical abstractions for several significant optimization problems in machine learning, combinatorial optimization, computer vision and statistical physics have intrinsic similarities. We propose a…

We propose a novel benchmarking methodology for graph neural networks (GNNs) based on the graph alignment problem, a combinatorial optimization task that generalizes graph isomorphism by aligning two unlabeled graphs to maximize overlapping…

We consider the task of estimating a high-dimensional directed acyclic graph, given observations from a linear structural equation model with arbitrary noise distribution. By exploiting properties of common random graphs, we develop a new…

The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the…

In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on…

Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several…

Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…

We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure…

This contribution presents a hierarchical multigrid approach for the solution of large-scale finite cell problems on both uniform grids and multi-level hp-discretizations. The proposed scheme leverages the hierarchical nature of the basis…

Neural ordinary differential equations (NODEs) are geometric deep learning models based on dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching…

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…

Folding is emerging as a promising manufacturing process to transform flat materials into functional structures, offering efficiency by reducing the need for welding, gluing, and molding, while minimizing waste and enabling automation.…

Linear diagrams are an effective way to visualize set-based data by representing elements as columns and sets as rows with one or more horizontal line segments, whose vertical overlaps with other rows indicate set intersections and their…

We examine applications of polynomial Lie algebras $sl_{pd}(2)$ to solve physical tasks in $G_{inv}$-invariant models of coupled subsystems in quantum physics. A general operator formalism is given to solve spectral problems using…