Hyperbolic Methods Research Articles

Euclidean and Poincaré space ensemble Xgboost

The Hyperbolic space has garnered attention for its unique properties and efficient representation of hierarchical structures. Recent studies have explored hyperbolic alternatives to hyperplane-based classifiers, such as logistic regression and support vector machines. Hyperbolic methods have even been fused into random forests by constructing data splits with horosphere, which proved effective for hyperbolic datasets. However, the existing incorporation of the horosphere leads to substantial computation time, diverting attention from its application on most datasets. Against this backdrop, we introduce an extension of Xgboost, a renowned machine learning (ML) algorithm to hyperbolic space, denoted as PXgboost. This extension involves a redefinition of the node split concept using the Riemannian gradient and Riemannian Hessian. Our findings unveil the promising performance of PXgboost compared to the algorithms in the literature through comprehensive experiments conducted on 64 datasets from the UCI ML repository and 8 datasets from WordNet by fusing both their Euclidean and hyperbolic-transformed (hyperbolic UCI) representations. Furthermore, our findings suggest that the Euclidean metric-based classifier performs well even on hyperbolic data. Building upon the above finding, we propose a space fusion classifier called, EPboost. It harmonizes data processing across various spaces and integrates probability outcomes for predictive analysis. In our comparative analysis involving 19 algorithms on the UCI dataset, our EPboost outperforms others in most cases, underscoring its efficacy and potential significance in diverse ML applications. This research marks a step forward in harnessing hyperbolic geometry for ML tasks and showcases its potential to enhance algorithmic efficacy.

Deep hyperbolic convolutional model for knowledge graph embedding

Recent advancements in knowledge graph embedding have enabled the representation of entities and relations in continuous vector spaces. Performing link prediction on incomplete knowledge graphs using these embeddings has emerged as a challenging task, drawing considerable research interest. Knowledge graphs in the real world typically exhibit a natural hierarchical structure. Hyperbolic embedding methods have shown promising results in modeling hierarchical data, especially in low-dimensional representations. However, existing hyperbolic models for knowledge graph embedding are generally shallow, resulting in limited expressiveness. Additionally, some deep hyperbolic models exist but are often constrained to fixed curvature spaces, which may not optimally capture varying hierarchical structures. In this work, we propose DeER, a novel multi-layer hyperbolic model that leverages a trainable curvature space, marking an advancement in the depth and adaptability of hyperbolic modeling for knowledge graph embeddings. We specifically design a hyperbolic convolutional neural network to learn the heterogeneous interactions between entities and relations within this adaptable hyperbolic space. Additionally, we introduce a hyperbolic feedforward neural network to transform hyperbolic features across multiple layers, enhancing the model’s ability to capture complex hierarchical relationships. Experimental results on three benchmark datasets demonstrate that DeER significantly enhances expressiveness and hierarchical modeling performance when compared to both shallow and other deep baseline approaches.

Performance improvement evaluation of latent heat energy storage units using improved bi-objective topology optimization method

Latent heat thermal energy storage (LHTES) based on phase change materials is one of the key technologies to improve energy utilization efficiency and alleviate the mismatch between energy supply and demand. Heat storage capacity and charging/discharging rate are two core factors that determine the comprehensive performance of LHTES units. So a bi-objective transient topology optimization(TO) model is established to freely evolve the optimal structure and volume fraction of LHTES fins to maximize the heat transfer rate based on ensuring high storage capacity. The bi-objective optimization function coupling storage capacity and heat transfer rate is constructed by normalized SAW model. The TO model is formulated using variable density-based design variables of grid cells in physical governing equations, objective functions and constraints. Helmholtz PDE filtering and hyperbolic tangent projection methods are employed to achieve high-precision optimization solutions. Complex transient multi-physics field and bi-objective iterative calculations are performed using the transient adjoint-based sensitivity analysis method and GCMMA algorithm. Results demonstrate that the physical field dynamics response of the design variables at each iteration can more clearly exhibit the evolutionary direction and mechanism of seeking optimal structures. The Pareto frontier indicates TO model responds significantly to the changes in the bi-objective weight ratio ω1:ω2, which also reveals the physical mechanism of the trade-off game among fin topology formation, state parameters distribution and objective changes. Optimal configurations and its performance parameters change regularly with ω1:ω2 decreasing, and has an demonstrable performance mutations occurring at ω1 = 0.65(ω2 = 0.35). The scattered distribution and uniform multi-branch structure of TO fins are key factors in regulating the physical field state. Therefore, TO-fins LHTES exhibits extremely high heat transfer rate, and the charging/discharging time of traditional trapezoid fin LHTES is approximately three times longer than that of TO-fins LHTES under ω1:ω2 = 0.65:0.35.

An investigation of Fokas system using two new modifications for the trigonometric and hyperbolic trigonometric function methods

In this work, two new adaptations for the trigonometric and hyperbolic trigonometric function approaches have been presented. These two modifications, entitled modified extended rational sin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sin$$\\end{document}–cos\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cos$$\\end{document} function technique and modified extended rational sinh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sinh$$\\end{document}–cosh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cosh$$\\end{document} function method, have been applied for the first time to the Fokas system that represents the nonlinear pulse propagation in monomode fiber optics. We intend to produce innovative, explicit traveling waves, solitons, and periodic wave solutions. These achieved outcomes are presented in the form of exponential functions, trigonometric hyperbolic functions, and combination constructions of the exponential functions along with the trigonometric and hyperbolic trigonometric functions. The obtained solutions reveal significant features of the physical phenomenon and are new. The investigated model incorporates the notions of dispersion, transverse diffusion, degree of dispersion, nonlinear pairing, nonlinear immersion, and the force of the nonlinear interaction among the two components of the system. For the most accurate visual evaluation of the physical importance and dynamic properties, we have presented the findings in a variety of plots, which involve two- and three-dimensional representations. One or more elements in our research that are unique, such as newly modified methodologies, is a new observation that leads researchers to invest in new solutions.

Lorentz equivariant model for knowledge-enhanced hyperbolic collaborative filtering

Introducing prior auxiliary information from the knowledge graph (KG) to assist the user–item graph can improve the comprehensive performance of the recommender system. Many recent studies have shown that the ensemble properties of hyperbolic spaces fit the scale-free and hierarchical characteristics exhibited in the above two types of graphs well. Therefore, hyperbolic-based recommender systems have achieved a series of outstanding performances. However, in existing hyperbolic methods, equivariance is not considered. Thus, they cannot generalize symmetric features under given transformations, which seriously limits the capability of the model. Moreover, they cannot balance preserving the heterogeneity and mining the high-order entity information to users across two graphs. To fill these gaps, we propose a rigorous Lorentz group equivariant knowledge-enhanced collaborative filtering (LECF) model. Innovatively, we jointly update the attribute embeddings (containing the high-order entity signals from the KG) and hyperbolic embeddings (the distance between hyperbolic embeddings reveals the recommendation tendency) via the LECF layer with Lorentz Equivariant Transformation. Moreover, we propose Hyperbolic Sparse Attention Mechanism to sample the most informative neighboring nodes. Lorentz equivariance is strictly maintained throughout the entire model, and enforcing equivariance is proven necessary experimentally. Extensive experiments on three real-world datasets demonstrate that the proposed LECF remarkably outperforms state-of-the-art methods.

Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives

In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

Disentangled representation learning for collaborative filtering based on hyperbolic geometry

In the realm of recommender systems, the exploration of hyperbolic geometry-based embeddings for users and items has emerged as a promising avenue, particularly in the context of collaborative filtering through graph convolution networks. Despite the advancements in this domain, there are two significant questions have received limited attention: (i) Most hyperbolic geometry-based methods ignore the influence of different users’ intents on the preferences of historical behaviors; (ii) They usually learn high-dimensional embeddings akin to Euclidean geometry-based methods, without taking good advantage of the capacity of hyperbolic spaces. To tackle these limitations head-on, we propose a novel method called Disentangled Hyperbolic Collaborative Filtering (DHCF). DHCF learns multiple embeddings in different low-dimensional hyperbolic spaces, enabling the disentanglement of users’ intents and the separate modeling of user and item representations. Moreover, we design an intent-aware graph reconstruction module, which adaptively allocates intent-aware relation strengths to build a dynamic interaction graph. Additionally, this module reduces the risk of oversmoothing in low-dimension spaces, facilitating the stacking of multiple graph aggregation layers. To the best of our knowledge, our method achieves competitive performance compared to state-of-the-art approaches.

Theoretical understanding of reaction and kinetics in the reduction of the bromate anion to bromine on a rotating disk electrode

The reduction of the bromate anion to bromine on a rotating disk electrode (RDE) at steady-state conditions is discussed. This model is based on a system of nonlinear equations. The most crucial aspect of this investigation is finding the analytical expression of the concentration of reagents by solving the coupled second-order nonlinear equation using hyperbolic function methods. The effects of the kinetic parameters on concentration and current are explored in the unitizing graph and tables. The analytical results are validated in numerical results using MATLAB software. The effects of the parameters such as diffusion layer thickness, rotation rate, rate constants and the ratio of concentration of bromate anion and volume of concentration of bromine ions on current are discussed.

Exact traveling wave solutions for (2+1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods

Abstract In this research, the extended rational sinh-cosh method and the modified extended tanh-function method for mathematically constructing traveling wave solutions to the (2+1)-dimensional integro-differential Konopelchenko-Dubrovsky evolution equation are successfully employed to obtain specific appropriate solutions for the first time. A traveling wave transformation was utilized to turn the provided model into a third-order nonlinear ordinary differential equation. Solitary and periodic wave solutions for the model under investigation are obtained in terms of various complex hyperbolic trigonometric and rational functions. Several of the aforementioned solutions have been represented by two- and three-dimensional graphics with appropriate arbitrary parameters to highlight their physical implications. Two-dimensional graphs have presented the influence of time evolution on the solution’s structures.

Elastic wave propagation in thick-walled hollow cylinders for damage localization through inner surface sensing

Thick-walled hollow cylinders (TWHCs) are widely used in engineering structures and transportation systems, exemplified by train axles. The real-time and online health monitoring of such structures is crucial to ensure their structural integrity and operational safety. While elastic-wave-based structural health monitoring (SHM) shows promise, the development of feasible methods strongly relies on a good understanding and exploitation of the wave propagation properties and their interaction with structural defects. TWHCs usually bear multiple wave modes, which is a less investigated and explored topic as compared with thin-walled structures. This work examines this issue and proposes a dedicated damage localization strategy by using the selected waves captured on the inner surface of a TWHC. It is shown that, alongside the quasi-surface-waves on the outer surface, longitudinal waves converted from the thickness-through shear bulk waves are generated to propagate along the inner surface. Their propagation characteristics are exploited for damage localization based on hyperbolic loci methods through inner surface sensing. Numerical studies are conducted to validate the method and assess different transducer configurations, alongside experimental verifications on a benchmark TWHC containing a notch-type defect. Studies provide guidance on damage detection in TWHCs and sensor network design.

Strain-minimizing hyperbolic network embeddings with landmarks

Abstract We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few ‘landmark nodes’. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in $d$-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just $d+1$ landmarks. We also show that L-hydra solves a two-stage strain-minimization problem, similar to our previous (unlandmarked) method ‘hydra’. Testing on real network data, we show that L-hydra is an order of magnitude faster than the existing hyperbolic embedding methods and scales linearly in the number of nodes. While the embedding error of L-hydra is higher than the error of the existing methods, we introduce an extension, L-hydra+, which outperforms the existing methods in both runtime and embedding quality.

Topology Optimization Design of Micro-Channel Heat Sink by Considering the Coupling of Fluid-Solid and Heat Transfer

To investigate the effect of the target weight coefficient on the structure design of the micro-channel heat sink, an innovative method for the topology optimization design of micro-channel structures with different bifurcation angles is adopted. In this study, the improved interpolation function, density filtering, and hyperbolic tangent projection methods are adopted to obtain a clear topological structure of the micro-channel heat sink. The heat transfer of the micro-channel heat sink under different bifurcation angles is compared. At the same time, the influence of the two different objective functions, heat transfer, and flow energy consumption, is analyzed in the topology optimization of micro-channel heat sinks. The results show that when the bifurcation angle is 135°, both the heat transfer and the average outlet temperature of the micro-channel heat sink obtain the maximum value, and the heat transfer effect is the best. With the increase of the heat transfer weighting coefficient, the distribution of solid heat sources in the main channel increases, and the refinement of the branch channels also increases. On the other hand, although the heat transfer effect of the micro-channel heat sink is the best, the corresponding flow energy consumption is larger.

Evaluation of Asaoka and Hyperbolic Methods for Settlement Prediction of Vacuum Preloading Combined with Prefabricated Vertical Drains in Soft Ground Treatment

This study evaluated the use of the Asaoka and hyperbolic methods to estimate the ultimate settlement of soft ground treated by vacuum preloading combined with prefabricated vertical drains. For this aim, a large-scale physical laboratory model was constructed. The model was a reinforced-tempered glass box containing a soil mass with dimensions of 2.0 × 1.0 × 1.2 m (length × width × depth). Physical models of this scale for the same purpose are rare in the literature. The soil was taken from a typical coastal region in Dinh Vu Hai Phong, Vietnam. The surface settlement near and between the two drains was measured right after the vacuum preloading started. Important properties of the soil were tested to evaluate the effectiveness of the treatment method. The measured settlement was used in the Asaoka and hyperbolic methods to predict the potential ultimate settlement. The results showed the superiority of the vacuum consolidation approach in improving fundamental engineering properties of soft soil. Furthermore, the ultimate settlement predicted by both methods showed a good agreement with the measured value, proving that the Asaoka and hyperbolic methods are suitable for the estimation of the ultimate settlement of soft soil treated with vacuum consolidation.

Gaussian Decline Curve Analysis of Hydraulically Fractured Wells in Shale Plays: Examples from HFTS-1 (Hydraulic Fracture Test Site-1, Midland Basin, West Texas)

The present study shows how new Gaussian solutions of the pressure diffusion equation can be applied to model the pressure depletion of reservoirs produced with hydraulically multi-fractured well systems. Three practical application modes are discussed: (1) Gaussian decline curve analysis (DCA), (2) Gaussian pressure-transient analysis (PTA) and (3) Gaussian reservoir models (GRMs). The Gaussian DCA is a new history matching tool for production forecasting, which uses only one matching parameter and therefore is more practical than hyperbolic DCA methods. The Gaussian DCA was compared with the traditional Arps DCA through production analysis of 11 wells in the Wolfcamp Formation at Hydraulic Fracture Test Site-1 (HFTS-1). The hydraulic diffusivity of the reservoir region drained by the well system can be accurately estimated based on Gaussian DCA matches. Next, Gaussian PTA was used to infer the variation in effective fracture half-length of the hydraulic fractures in the HFTS-1 wells. Also included in this study is a brief example of how the full GRM solution can accurately track the fluid flow-paths in a reservoir and predict the consequent production rates of hydraulically fractured well systems. The GRM can model reservoir depletion and the associated well rates for single parent wells as well as for arrays of multiple parent–parent and parent–child wells.

Diverse image enhancer for complex underexposed image

The visual appearance of images changes in line with varying environmental light conditions. Adjusting the exposure of various distorted images is a highly complex process. Previous approaches have addressed this issue from different viewpoints and attained remarkable progress. However, they either failed to achieve visually pleasing results or were suitable for a single class of images (e.g., underexposed or nonuniform images). To fully consider the exposure in various distorted images, we proposed a diverse image enhancement model that improved the brightness and contrast, processed the colors, and eliminated the hazy effect. Accordingly, an input red green blue color image was transformed into a hue, saturation, value color image. The V component was inverted and enhanced using three steps. In the first step, the hyperbolic and statistical methods were applied, and then their results were combined using an adjusted logarithmic methodology. This method properly adjusted the high-contrast and low-contrast impact while preserving the vital image information. In the second step, the output of the first step was inverted back and fed into a complete optimization algorithm to estimate the illumination map. Then, the exposure ratio map was estimated using an illumination map, which was adjusted using the camera response function. In the third step, a nonlinear stretching function was introduced to control brightness and contrast. For instance, a lower value of α yielded maximum stretching, and a higher value of α eliminated haze in the image to a great extent. Finally, an empirical evaluation and comparison of the most recent state-of-the-art approaches on eight datasets revealed that the proposed model efficiently addressed the exposure in various degraded images.

Reaction-diffusion in a packed-bed reactors: Enzymatic isomerization with Michaelis-Menten Kinetics

A theoretical model for the enzyme catalysed isomerisation reaction in a packed-bed reactors made up of microporous particles is discussed. The system is described in terms of finite-range Fickian diffusion and non linear Michaelis-Menten reaction kinetics to produce a non linear reaction/diffusion equation which is solved both numerically by MATLAB and analytically by hyperbolic function methods (HFM). Approximate analytical expressions of the substrate concentration within the microparticle, the related flux and the effectiveness factor for a wide range of values of pore-level Thiele modulus and kinetic parameters are reported using these simple, and robust methods. A satisfactory agreement is obtained between the estimated analytical solution and the numerical simulation.

Knowledge embedding via hyperbolic skipped graph convolutional networks

With the aim of constructing a low-dimensional representation space, Hyperbolic Knowledge Embeddings have gradually become a hot spot in various information retrieval and machine learning tasks. However, most of the existing Hyperbolic knowledge embedding methods focus on the shallow embedding, and often ignore the network structure characteristics (e.g., hierarchy) of Knowledge Graphs. Therefore, this paper designs a novel Hyperbolic Skipped Knowledge Graph Convolutional Network, HSKGCN, to improve link prediction accuracy with low embedding dimension requirements. Firstly, the model is designed based on the hyperbolic geometric operations on the Poincaré ball, which can effectively utilize the characteristics of the hyperbolic geometry (e.g., Poincaré ball) to capture the hierarchy of Knowledge Graphs; Secondly, each single-layer convolutional layer introduces the feature aggregation weight, which ensures the reasonable distribution of node features during the aggregation process; In addition, the skip-connection mechanism is applied to HSKGCN to weaken the information loss caused by the stacked of the graph convolutional layers; Finally, we evaluate HSKGCN on benchmark datasets, WN18RR and FB15k-237. Experiments show that HSKGCN achieves substantial improvements against state-of-the-art models on the 32-dimensional embedding task, and the results of different relations on WN18RR show graphs similar with tree topology can performer better.

Accurate Tree Roots Positioning and Sizing Over Undulated Ground Surfaces by Common Offset GPR Measurements

Tree roots detection is a popular application of the Ground-penetrating radar (GPR). Normally, the ground surface above the tree roots is assumed to be flat, and standard processing methods based on hyperbolic fitting are applied to the hyperbolae reflection patterns of tree roots for detection purposes. When the surface of the land is undulating (not flat), these typical hyperbolic fitting methods becomes inaccurate. This is because, the reflection patterns change with the uneven ground surfaces. When the soil surface is not flat, it is inaccurate to use the peak point of an asymmetric reflection pattern to identify the depth and horizontal position of the underground target. The reflection patterns of the complex shapes due to extreme surface variations results in analysis difficulties. Furthermore, when multiple objects are buried under an undulating ground, it is hard to judge their relative positions based on a B-scan that assumes a flat ground. In this paper, a roots fitting method based on electromagnetic waves (EM) travel time analysis is proposed to take into consideration the realistic undulating ground surface. A wheel-based (WB) GPR and an antenna-height-fixed (AHF) GPR System are presented, and their corresponding fitting models are proposed. The effectiveness of the proposed method is demonstrated and validated through numerical examples and field experiments.

Systematic comparison of graph embedding methods in practical tasks.

Network embedding techniques aim to represent structural properties of graphs in geometric space. Those representations are considered useful in downstream tasks such as link prediction and clustering. However, the number of graph embedding methods available on the market is large, and practitioners face the nontrivial choice of selecting the proper approach for a given application. The present work attempts to close this gap of knowledge through a systematic comparison of 11 different methods for graph embedding. We consider methods for embedding networks in the hyperbolic and Euclidean metric spaces, as well as nonmetric community-based embedding methods. We apply these methods to embed more than 100 real-world and synthetic networks. Three common downstream tasks - mapping accuracy, greedy routing, and link prediction - are considered to evaluate the quality of the various embedding methods. Our results show that some Euclidean embedding methods excel in greedy routing. As for link prediction, community-based and hyperbolic embedding methods yield an overall performance that is superior to that of Euclidean-space-based approaches. We compare the running time for different methods and further analyze the impact of different network characteristics such as degree distribution, modularity, and clustering coefficients on the quality of the embedding results. We release our evaluation framework to provide a standardized benchmark for arbitrary embedding methods.

On Certain Applications of the Hyperbolic Heat Transfer Equation and Methods for Its Solution

When creating new technological processes based on the use of high-intensity energy flows, it is necessary to take into account the finite speed of heat transfer. This can be done using the hyperbolic heat transfer equation obtained by A. V. Lykov within the framework of nonequilibrium phenomenological thermodynamics as a consequence of a generalization of the Fourier law for flows and the equation of thermal balance. In previous works, V. N. Khankhasaev modeled the process of switching off an electric arc in a gas flow using this equation. In this paper, we present a mathematical model of this process including the period of stable burning of an electric arc until the shutdown moment, which consists of the replacement of the strongly hyperbolic heat transfer equation by a hyperbolicparabolic equation. For the mixed heat transfer equation obtained, we state certain boundary-value problems, solve them by numerical algorithms, and obtain temperature fields that are consistent with the available experimental data.