Euclidean Domains Research Articles

Pseudo-Graph Neural Networks On Ordinary Differential Equations

In this paper, we extend the idea of continuous-depth models to pseudo graphs and present pseudo graph ordinary differential equations (PGODE), which are inspired by the neural ordinary differential equation (NODE) for data in the Euclidean domain. All existing graph networks have discrete depth. A pseudo graph neural network (PGNN) is used to parameterize the derivative of hidden node states, and the output states are the solution to this ordinary differential equation (ODE). A memory-efficient framework with precise gradient estimates is then proposed for free-form ODEs. We also introduce the framework of continuous–depth pseudo graph neural networks (PGNNs) on ODE by blending discrete structures and differential equations

Quantum trees which maximize higher eigenvalues are unbalanced

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the k k -th positive eigenvalue is the star graph with three edges of lengths 2 k − 1 2 k - 1 , 1 1 and 1 1 . This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape—an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.

No-slip billiards with particles of variable mass distribution.

Astute variations in the geometry of mathematical billiard tables have been and continue to be a source of understanding their wide range of dynamical behaviors, from regular to chaotic. Viewing standard specular billiards in the broader setting of no-slip (or rough) collisions, we show that an equally rich spectrum of dynamics can be called forth by varying the mass distribution of the colliding particle. We look at three two-parameter families of billiards varying both the geometry of the table and the particle, including as special cases examples of standard billiards demonstrating dynamics from integrable to chaotic, and show that markedly divergent dynamics may arise by changing only the mass distribution. Furthermore, for certain parameters, billiards emerge, which display unusual dynamics, including examples of full measure periodic billiards, conjectured to be nonexistent for the standard billiards in Euclidean domains.

Guest Editorial: Non-Euclidean Machine Learning

Over the past decade, deep learning has had a revolutionary impact on a broad range of fields such as computer vision and image processing, computational photography, medical imaging and speech and language analysis and synthesis etc. Deep learning technologies are estimated to have added billions in business value, created new markets, and transformed entire industrial segments. Most of today’s successful deep learning methods such as Convolutional Neural Networks (CNNs) rely on classical signal processing models that limit their applicability to data with underlying Euclidean grid-like structure, e.g., images or acoustic signals. Yet, many applications deal with non-Euclidean (graph- or manifold-structured) data. For example, in social network analysis the users and their attributes are generally modeled as signals on the vertices of graphs. In biology protein-to-protein interactions are modeled as graphs. In computer vision & graphics 3D objects are modeled as meshes or point clouds. Furthermore, a graph representation is a very natural way to describe interactions between objects or signals. The classical deep learning paradigm on Euclidean domains falls short in providing appropriate tools for such kind of data. Until recently, the lack of deep learning models capable of correctly dealing with non-Euclidean data has been a major obstacle in these fields. This special section addresses the need to bring together leading efforts in non-Euclidean deep learning across all communities. From the papers that the special received twelve were selected for publication. The selected papers can naturally fall in three distinct categories: (a) methodologies that advance machine learning on data that are represented as graphs, (b) methodologies that advance machine learning on manifold-valued data, and (c) applications of machine learning methodologies on non-Euclidean spaces in computer vision and medical imaging. We briefly review the accepted papers in each of the groups.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

A Review of Graph Neural Networks and Their Applications in Power Systems

Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data in these tasks are typically represented in Euclidean domains. Nevertheless, there is an increasing number of applications in power systems, where data are collected from non-Euclidean domains and represented as graph-structured data with high-dimensional features and interdependency among nodes. The complexity of graph-structured data has brought significant challenges to the existing deep neural networks defined in Euclidean domains. Recently, many publications generalizing deep neural networks for graph-structured data in power systems have emerged. In this paper, a comprehensive overview of graph neural networks (GNNs) in power systems is proposed. Specifically, several classical paradigms of GNN structures, e. g., graph convolutional networks, are summarized. Key applications in power systems such as fault scenario application, time-series prediction, power flow calculation, and data generation are reviewed in detail. Further-more, main issues and some research trends about the applications of GNNs in power systems are discussed.

Some rigidity results for minimal graphs over unbounded Euclidean domains

<p style='text-indent:20px;'>We prove some new rigidity results for minimal graphs over unbounded Euclidean domains. In particular we prove that a positive minimal graph over an open affine half-space, and under the homogenous Dirichlet boundary condition, must be an affine function.</p>

Multiscale regression on unknown manifolds

<abstract><p>We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.</p></abstract>

Uniformization of metric surfaces using isothermal coordinates


 We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.

Some applications of heat flow to Laplace eigenfunctions

We consider mass concentration properties of Laplace eigenfunctions that is, smooth functions satisfying the equation on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central “thick” part and “thin” elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.

Lifespan estimates for 2-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if 1<p≤pc(2), the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for 1<p≤2. It is interesting to see that the logarithmic increase of the harmonic function in 2-D has no influence to the estimate of the upper bound of the lifespan for 2<p≤pc(2). One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simply connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

Abstract This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets K∪M in a complex manifold X, where K is a compact holomorphically convex set in an open Stein neighbourhood, M is an embedded Stein submanifold, and K∩M is compact O(M)-convex. We prove a Docquier–Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of K∪M, biholomorphic to domains in Cn with n=dim⁡X, such that M is mapped onto a closed complex submanifold of Cn.

Modeling spatial correlation that grows on trees, with a stream network application

Spatial data on a network, like spatial data on a Euclidean domain, may exhibit nonstationarity. This article develops two classes of nonstationary models for continuously indexed data on directed tree networks, such as stream networks, that are adaptations of models used previously for nonstationary temporal or spatial data on Euclidean domains. These classes, called elastic models and spatially varying moving average models, allow the spatial dependence between observations at sites any fixed distance apart to grow monotonically as one moves either up or down the network. The process variance, or components thereof, may also be allowed to grow monotonically. An example of trout density data from a stream network in Wyoming, USA indicates that the proposed nonstationary models fit those data much better than their existing stationary or quasi-stationary counterparts.

Graph representation learning in bioinformatics: trends, methods and applications.

Graph is a natural data structure for describing complex systems, which contains a set of objects and relationships. Ubiquitous real-life biomedical problems can be modeled as graph analytics tasks. Machine learning, especially deep learning, succeeds in vast bioinformatics scenarios with data represented in Euclidean domain. However, rich relational information between biological elements is retained in the non-Euclidean biomedical graphs, which is not learning friendly to classic machine learning methods. Graph representation learning aims to embed graph into a low-dimensional space while preserving graph topology and node properties. It bridges biomedical graphs and modern machine learning methods and has recently raised widespread interest in both machine learning and bioinformatics communities. In this work, we summarize the advances of graph representation learning and its representative applications in bioinformatics. To provide a comprehensive and structured analysis and perspective, we first categorize and analyze both graph embedding methods (homogeneous graph embedding, heterogeneous graph embedding, attribute graph embedding) and graph neural networks. Furthermore, we summarize their representative applications from molecular level to genomics, pharmaceutical and healthcare systems level. Moreover, we provide open resource platforms and libraries for implementing these graph representation learning methods and discuss the challenges and opportunities of graph representation learning in bioinformatics. This work provides a comprehensive survey of emerging graph representation learning algorithms and their applications in bioinformatics. It is anticipated that it could bring valuable insights for researchers to contribute their knowledge to graph representation learning and future-oriented bioinformatics studies.

Local pointwise second derivative estimates for strong solutions to the sigma _k-Yamabe equation on Euclidean domains

We prove local pointwise second derivative estimates for positive W^{2,p} solutions to the sigma _k-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.

Quasiregular curves: Hölder continuity and higher integrability

We show that a K-quasiregular $$\omega$$ -curve from a Euclidean domain to a Euclidean space with respect to a covector $$\omega$$ is locally $$(1/K)(\Vert \omega \Vert /|\omega |_{\ell _1})$$ -Hölder continuous. We also show that quasiregular curves enjoy higher integrability.

Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory

In the 1960s, Almgren [3, 4] initiated a program to find minimal hypersurfaces in Riemannian manifolds using min-max method. This program was largely advanced by Pitts [34] and Schoen–Simon [37] in the 1980s when the manifold is compact without boundary. In this paper, we finish this program for general compact manifold with non-empty boundary. As a corollary, we establish the existence of a smooth embedded minimal hypersurface with non-empty free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves [31] shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many properly embedded minimal hypersurfaces with non-empty free boundary.

Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

Abstract In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.

Semi-Supervised Classification via Hypergraph Convolutional Extreme Learning Machine

Extreme Learning Machine (ELM) is characterized by simplicity, generalization ability, and computational efficiency. However, previous ELMs fail to consider the inherent high-order relationship among data points, resulting in being powerless on structured data and poor robustness on noise data. This paper presents a novel semi-supervised ELM, termed Hypergraph Convolutional ELM (HGCELM), based on using hypergraph convolution to extend ELM into the non-Euclidean domain. The method inherits all the advantages from ELM, and consists of a random hypergraph convolutional layer followed by a hypergraph convolutional regression layer, enabling it to model complex intraclass variations. We show that the traditional ELM is a special case of the HGCELM model in the regular Euclidean domain. Extensive experimental results show that HGCELM remarkably outperforms eight competitive methods on 26 classification benchmarks.