Geometric Representation Research Articles

Pursuing Extreme Descriptive Power and Generality in Chemical Bond Theories: A Method to Decipher "Interatomic Genomes" from Interatomic Electron Structures.

The description and analysis of chemical bonds have been difficult following the popularization of electronic structure calculations. Although many attempts have been made from the perspective of electronic structure, the sheer volume of information in the electronic structure has left contemporary chemical bond analysis methods grappling with an inescapable "Trilemma" where the model briefness, generality, and descriptiveness (descriptive power) cannot be obtained simultaneously. To push the generality and descriptiveness to their extremes, herein a general machine learning-based framework is introduced to compact chemical bonds into a detailed residue-by-residue "genome" with matched encoding/decoding tools. The framework fuses the quantum mechanical aspects, auto feature extraction, nanostructures and/or simulations, and generative models. The encoded genomes are information-dense and decodable, where 100% generality is guaranteed. The descriptiveness of genomes appears to be broader than most known models. As a proof of concept, the realization presented in this work compacts the complete information regarding two critical chemical bonds in thiolate-protected gold nanoclusters, the S-Au and Au-Au bonds, from a Bosonic-Fermionic character perspective into 8-valued genomes. The machine learning component is trained based on 26,528 density functional theory simulated electron localization function images. With an exploration of the space span for the genome, bond polarization, hybridization, intrusion of other atoms, alignments, crystal orientation, atomic motions, and more details are observed. Furthermore, it has emerged from extensive generation tests that molecules and solids can be integrated in such a concise manner than is typically achieved with purely geometric representations. To showcase the intraclass complexity of S-Au and Au-Au bonds visually, a roadmap is plotted by summarizing and correlating the similarities of 8-value-genomes. Furthermore, genomes can be associated with realistic indices easily with a simple multilayer perception architecture as a simple calculating tool. Besides, there are 3 sets of applications, including a set of chemisorption, a set of molecular dynamical analysis, and a set of ultrafast processes, showcasing the interpretability potentials of interatomic genomes in the geometric structures, kinetic properties, and vibration characteristics of molecular systems. As the framework rose to the challenge of nanoclusters from a complicated mesoscopic family of material, the displayed generality and comprehensiveness indicate that the model may "understand" chemical bonds in a machine's way.

Partial density of states representation for accurate deep neural network predictions of X-ray spectra.

The performance of a machine learning (ML) algorithm for chemistry is highly contingent upon the architect's choice of input representation. This work introduces the partial density of states (p-DOS) descriptor: a novel, quantum-inspired structural representation which encodes relevant electronic information for machine learning models seeking to simulate X-ray spectroscopy. p-DOS uses a minimal basis set in conjunction with a guess (non-optimised) electronic configuration to extract and then discretise the density of states (DOS) of the absorbing atom to form the input vector. We demonstrate that while the electronically-focused p-DOS performs well in isolation, optimal performance is achieved when supplemented with nuclear structural information imparted via a geometric representation. p-DOS provides a description of the key electronic properties of a system which is not only concise and computationally efficient, but also independent of molecular size or choice of basis set. It can be rapidly generated, facilitating its application with large training sets. Its performance is demonstrated using a wide variety of examples at the sulphur K-edge, including the prediction of ultrafast X-ray spectroscopic signal associated with photoexcited 2(5H)-thiophenone. These results highlight the potential for ML models developed using p-DOS to contribute to the interpretation and prediction of experimental results e.g. in operando measurements of batteries and/or catalysts and femtosecond time-resolved studies, especially those made possible by emergent cutting-edge technologies, especially X-ray free electron lasers.

A comprehensive method for error separation in hydrological modelling

AbstractEvaluation metrics play a pivotal role in the calibration process of hydrological models, serving as objective functions that directly influence the final values of model parameters and significantly affect users' perceptions of model performance. However, the choice and interpretation of evaluation metrics are subjective; therefore, this study provides a more objective framework for assessing model performance. This paper initially explored the applicability of various commonly used evaluation metrics, providing an overview of their limitations. Following this, we decomposed errors by analysing their physical significance and geometric representation in scatter plots, categorizing them into systematic and unsystematic errors. Through the decomposition and derivation of the Nash–Sutcliffe efficiency (NSE) formula, we established the quantitative relationship among various evaluation metrics. The soil and water assessment tool (SWAT) model was utilized to simulate monthly runoff in the Baishan basin (China), for the period 1994–2017, with NSE serving as the objective function for calibration. Our findings are consistent with previous studies, indicating that the model tends to slightly underestimate high flows while significantly overestimating low flows. Further analysis through error decomposition and the examination of relationships among various evaluation metrics revealed that unsystematic errors were dominant during the spring snowmelt runoff period, while systematic errors prevailed in the dry season. By evaluating the runoff series based on the magnitude of runoff or by categorizing it according to seasons and months, a more stringent assessment of the model's performance was achieved. These findings not only highlight the necessity for careful selection of evaluation metrics but also underscore the significance of our methodological advancements in enhancing hydrological model precision and reliability.

Prediction of mutation-induced protein stability changes based on the geometric representations learned by a self-supervised method

BackgroundThermostability is a fundamental property of proteins to maintain their biological functions. Predicting protein stability changes upon mutation is important for our understanding protein structure–function relationship, and is also of great interest in protein engineering and pharmaceutical design.ResultsHere we present mutDDG-SSM, a deep learning-based framework that uses the geometric representations encoded in protein structure to predict the mutation-induced protein stability changes. mutDDG-SSM consists of two parts: a graph attention network-based protein structural feature extractor that is trained with a self-supervised learning scheme using large-scale high-resolution protein structures, and an eXtreme Gradient Boosting model-based stability change predictor with an advantage of alleviating overfitting problem. The performance of mutDDG-SSM was tested on several widely-used independent datasets. Then, myoglobin and p53 were used as case studies to illustrate the effectiveness of the model in predicting protein stability changes upon mutations. Our results show that mutDDG-SSM achieved high performance in estimating the effects of mutations on protein stability. In addition, mutDDG-SSM exhibited good unbiasedness, where the prediction accuracy on the inverse mutations is as well as that on the direct mutations.ConclusionMeaningful features can be extracted from our pre-trained model to build downstream tasks and our model may serve as a valuable tool for protein engineering and drug design.

Using multiple, dynamically linked representations to develop representational competency and conceptual understanding of the earthquake cycle

Using computational methods to produce and interpret multiple scientific representations is now a common practice in many science disciplines. Research has shown students have difficulty in moving across, connecting, and sensemaking from multiple representations. There is a need to develop task-specific representational competencies for students to reason and conduct scientific investigations using multiple representations. In this study, we focus on three representational competencies: 1) linking between representations, 2) disciplinary sensemaking from multiple representations, and 3) conceptualizing domain-relevant content derived from multiple representations. We developed a block code-based computational modeling environment with three different representations and embedded it within an online activity for students to carry out investigations around the earthquake cycle. The three representations include a procedural representation of block codes, a geometric representation of land deformation build-up, and a graphical representation of deformation build-up over time. We examined the extent of students' representational competencies and which competencies are most correlated with students’ future performance in a computationally supported geoscience investigation. Results indicate that a majority of the 431 students showed at least some form of representational competence. However, a relatively small number of students showed sophisticated levels of linking, sensemaking, and conceptualizing from the representations. Five of seven representational competencies, the most prominent being code sensemaking (η2 = 0.053, p < 0.001), were significantly correlated to student performance on a summative geoscience investigation.

An open-source MATLAB toolbox for 3D block cutting and 3D mesh cutting in geotechnical engineering

This study presents a simple approach and its associated MATLAB toolbox for 3D block cutting and mesh cutting. The approach is suitable for the meshes of numerical methods including the Key Block Theory (KBT), the Discontinuous Deformation Analysis (DDA), the Numerical Manifold Method (NMM), and the cut Finite Element Method (Cut-FEM). The strategy is based on calculations on convex bodies. It uses two different forms of representation: the geometric representation which includes vertices and faces, and the algebraic representation which consists of inequalities. The cutting was implemented on the algebraic representation, and the resulting inequalities were converted into a geometric representation. The above strategy turned out to be robust and straightforward to execute, at the cost of a general body being regarded as a combination of convex bodies. The efficiency guarantee was considered through pre-checking algorithms. The source code was provided, as well as some simple examples.

Geometric Phase-Driven Scattering Evolutions.

Conventional approaches for scattering manipulations largely rely on the technique of field expansions into spherical harmonics (electromagnetic multipoles), which nevertheless is not only nongeneric (expansion coefficients depend on the origin position of the coordinate system) but also more descriptive than predictive. Here, we explore this classical topic from a different perspective of controlled excitations and interferences of quasinormal modes (QNMs) supported by the scattering system. Scattered waves are expanded into coherent additions of QNMs, among which the relative amplitudes and phases are crucial factors to architect for scattering manipulations. Relying on the electromagnetic reciprocity, we provide full geometric representations based on the Poincaré sphere for those factors, and discover the hidden geometric phase of QNMs that drives the scattering evolutions. Further synchronous exploitations of the incident polarization-dependent geometric phase and excitation amplitudes enable efficient manipulations of both scattering intensities and polarizations. Continuous geometric phase spanning 2π is directly manifest through scattering variations, even in the rather elementary configuration of an individual particle scattering waves of varying polarizations. We have essentially established a profoundly all-encompassing framework for the calculations of geometric phase in arbitrary scattering systems that are reciprocal. Our theoretical model will greatly broaden horizons of many disciplines not only in photonics but also in general wave physics where geometric phase is generic and ubiquitous.

On the Geometric Pattern Transformation (GPT) Properties of Unidimensional Signals

The Geometric Pattern Transformation (GPT) has several advantages of use concerning contemporary algorithms that have been duly studied in previous research. Regarding some of its properties, four different but complementary aspects of the GPT are presented in this work. After a brief review of the GPT concept, how tied data are manifested in data sets is shown, to obtain a symmetric representation of the GPT, a linear transformation is performed that regularizes the geometric representation of the GPT and the theoretical relationship between the GPT and the phase-state representation of 1D signals is analyzed and formalized, then the study of the forbidden pattern is easily revealed, obtaining a strong relationship with the stable and unstable fixed points of the logistic equation. Finally, the characterization of colored noises and the application in real world signals taken through experimental procedures is analyzed. With these results, in this work is proposed an advance in the potential applications of the GPT in an integral way in the processing and analysis of data series.

A parametric modeling method for 3D woven composites considering realistic meso-structural characteristics

The precision of mesoscale simulation for three-dimensional woven composites (3DWCs) is intricately linked to the fidelity of the geometric representation. This paper aims to present a novel parametric modeling approach for generating representative volume element (RVE) of the 3DWCs while considering its realistic meso-structural characteristics. The real architecture of 3DWCs is defined to consider the squeezed surface warp tow. Tow geometry is specified, accounting for torsion of tow cross-section and crimp of weft tow path. The RVEs of the composites are geometrically assembled via a specific translational symmetry. The tensile response of the composites with the novel geometry is scrutinized utilizing a progressive damage model, compared with that of the ideal geometry. Additionally, the impact of weft tow size on the tensile response of the composites is explored.

A fibre direction informed continuum damage model with enriched kinematics for modelling matrix cracking in composites

Continuum Damage Mechanics models are widely used for simulation of ply-level damage such as matrix cracking in composites. Although their ease of implementation within the finite element framework is appealing, several limitations exist such as a smeared representation of sharp cracks, mesh orientation bias of crack growth, inaccurate energy dissipation due to approximate definition of characteristic length etc. In this work, a new ‘semi-discrete’ continuum damage model is proposed for matrix cracking, that includes sharp crack kinematics. The mesh orientation bias is eliminated by a fibre-directed growth driving criterion. The crack characteristic length is exactly computed due to an accurate geometric representation of the crack surface. The model also addresses a well-known problem of spurious stress transfer across crack faces when deformation becomes large. Through several challenging benchmark problems, the model is shown to perform nearly as good as discrete crack models while being simpler and more robust.

A shape-supervised feature fusion U-Net for tubular structure segmentation

Accurate segmentation of tubular structures, such as blood vessels and bile ducts, is pivotal for clinical diagnosis and subsequent treatment. However, challenges arise from their unique structure and morphology. These structures are thin and elongated, and often exhibit complex branching patterns, making precise delineation difficult. Additionally, the intricate and often noisy backgrounds in medical images complicate the extraction of detailed and shape features. These challenges collectively hinder the effectiveness of automated segmentation methods. To address these issues, this paper introduces the shape-supervised feature fusion U-Net (SFU-Net), a novel approach that extends the traditional U-Net architecture. Our network incorporates a feature fusion module that effectively combines different semantic information in both the encoder and decoder paths, facilitating the capture of detailed geometric representations of tubular structures. Furthermore, a novel distance-based tubularity loss is designed to capture boundary information and supervise segmentation integrity by imposing shape constraints. We conducted comprehensive experiments on three tubular structure segmentation tasks in medical imaging, including the dilated biliary tree, hepatic vessel, and pulmonary artery. The experimental results illustrate that the proposed SFU-Net achieves the best performance in terms of accuracy and completeness, with Dice scores of 76.14%, 80.42%, and 83.79%, and HD95 values of 8.59 mm, 6.43 mm, and 4.31 mm across three datasets, respectively. Compared to traditional U-Net-based segmentation methods and other tubular segmentation approaches, our method exhibits superior performance, underscoring its effectiveness in medical image segmentation with tubular structures.

Interference of prototypical geometry representations in students’ construction of concepts and development of proofs

ABSTRACT This study investigates the impact of prototypical geometric representations on students’ ability to construct accurate geometric concepts and their subsequent performance in proving tasks related to those concepts. Ninety-one 11th-grade students participated in the study. We intentionally designed and utilized a questionnaire as the primary data-collection tool. We used the questionnaire items and findings as a basis for semi-structured interviews that we conducted with approximately 10% of the study participants. This allowed the integration of the quantitative and qualitative analyses, to provide a comprehensive understanding of the learning processes. Our analysis reveals that many Grade 11 students struggle with geometric concept formation due to an overreliance on prototypical examples, which in turn hinders their ability to construct proofs concerning the attributes of these concepts, highlighting the need for more diverse representations in geometry instruction.

Bilinear Distance Feature Network for Semantic Segmentation in PowerLine Corridor Point Clouds.

Semantic segmentation of target objects in power transmission line corridor point cloud scenes is a crucial step in powerline tree barrier detection. The massive quantity, disordered distribution, and non-uniformity of point clouds in power transmission line corridor scenes pose significant challenges for feature extraction. Previous studies have often overlooked the core utilization of spatial information, limiting the network's ability to understand complex geometric shapes. To overcome this limitation, this paper focuses on enhancing the deep expression of spatial geometric information in segmentation networks and proposes a method called BDF-Net to improve RandLA-Net. For each input 3D point cloud data, BDF-Net first encodes the relative coordinates and relative distance information into spatial geometric feature representations through the Spatial Information Encoding block to capture the local spatial structure of the point cloud data. Subsequently, the Bilinear Pooling block effectively combines the feature information of the point cloud with the spatial geometric representation by leveraging its bilinear interaction capability thus learning more discriminative local feature descriptors. The Global Feature Extraction block captures the global structure information in the point cloud data by using the ratio between the point position and the relative position, so as to enhance the semantic understanding ability of the network. In order to verify the performance of BDF-Net, this paper constructs a dataset, PPCD, for the point cloud scenario of transmission line corridors and conducts detailed experiments on it. The experimental results show that BDF-Net achieves significant performance improvements in various evaluation metrics, specifically achieving an OA of 97.16%, a mIoU of 77.48%, and a mAcc of 87.6%, which are 3.03%, 16.23%, and 18.44% higher than RandLA-Net, respectively. Moreover, comparisons with other state-of-the-art methods also verify the superiority of BDF-Net in point cloud semantic segmentation tasks.

A Study on the Steady and Dynamic Performance of Dry Gas Seals with Combined Superelliptical Grooves and Holes on the End Face

Background:: Enhancing the stability and leakage control of Dry Gas Seals (DGS) under high parameters has been a crucial research focus. The design of end-face groove structures and surface texture shapes has been an essential aspect of DGS studies aimed at improving performance. Objective:: The proposed end-face gas seal utilizes superelliptical grooves and holes to improve its performance, aiming to obtain a patent. The use of superelliptical curves allows for a more precise and efficient geometric representation, resulting in a better sealing effect. Methods:: A theoretical analysis model for the steady and dynamic behavior of the seal is established using the lubrication theory and perturbation methods. The study investigates the distribution patterns of gas film pressure on the sealing end-face and gas flow characteristics within the film. This approach provides a new perspective for understanding seal performance and offers a theoretical basis for optimizing seal design. Results:: The results indicate that the combined end-face structure with grooves and holes ensures good sealing stability and effectively enhances leakage control performance. By optimizing the design of the superelliptical groove and holes on the end face, the performance of DGS can be significantly improved. Conclusion:: Within the parameter range studied, better steady-state performance is achieved for θ=40~80°, v=1.3~1.4, u=1~2, β=0.6~0.7, and λ=1.0~1.5. In addition, better dynamic performance is observed for θ=80~120°, v=1.1~1.2, u=2~3, β=0.9~1.0, and λ=2.0~2.5.

Unravelling the mechanics of knitted fabrics through hierarchical geometric representation

Knitting interloops one-dimensional yarns into three-dimensional fabrics that exhibit behaviour beyond their constitutive materials. How extensibility and anisotropy emerge from the hierarchical organization of yarns into knitted fabrics has long been unresolved. We seek to unravel the mechanical roles of tensile mechanics, assembly and dynamics arising from the yarn level on fabric nonlinearity by developing a yarn-based dynamical model. This physically validated model captures the mechanical response of knitted fabrics, analogous to flexible metamaterials and biological fibre networks due to geometric nonlinearity within such hierarchical systems. Fabric anisotropy originates from observed yarn–yarn rearrangements during alignment dynamics and is topology-dependent. This yarn-based model also provides a design space of knitted fabrics to embed functionalities by varying geometric configuration and material property in instructed procedures compatible to machine manufacturing. Our hierarchical approach to build up a knitted fabric computationally modernizes an ancient craft and represents a first step towards mechanical programmability of knitted fabrics in wide engineering applications.

Non-conformance aspects of moulded composite materials and “corresponding” simulation models with 3D textile reinforcement

Composite materials with 3-dimensional (3D) reinforcement were manufactured and corresponding simulation models were created in parallel. The used simulation approach has earlier been shown to produce close to authentic geometrical representation of the yarn architecture in 3D reinforcement. It is shown that although the as-woven reinforcement pattern can be modelled quite reliably, significant distortion from the nominal fibre arrangement might take place later in manufacturing, primarily related to compression during moulding. Such effects have earlier received significant attention for composites with 2-dimensional reinforcement but not as much for their 3D counterparts. The yarns in the real and the simulated materials are studied and compared, and some of the discrepancies and the mechanisms behind are discussed. The distortions are partly attributed to the relatively sparse weave that allows yarns oriented in the through-thickness direction, in particular, to deviate from their original positions.

Optical edge-to-screw singularity state conversions

Optical singularity states, which significantly affect propagation properties of light in free space or optical medium, can be geometrically classified into screw and edge types. These different types of singularity states do not exhibit direct connection, being decoupled from each other in the absence of external perturbations. Here we demonstrate a novel optical process in which a higher-order edge singularity state initially nested in the propagating Gaussian light field gradually involves into a screw singularity with a new-born topological charge determined by order of the edge state. The considered edge state comprises an equal superposition of oppositely charged vortex and antivortex modes. We theoretically and experimentally realize this edge-to-screw conversion process by introducing intrinsic vortex–antivortex interaction. We also present a geometrical representation for mapping this dynamical process, based on the higher-order orbital Poincaré sphere. Within this framework, the edge-to-screw conversion is explained by a mapping of state evolution from the equator to the north or south pole of the Poincaré sphere. Our demonstration provides a novel approach for manipulating singularity state by the intrinsic vortex–antivortex interactions. The presented phenomenon can be also generalized to other wave systems such as matter wave, water wave, and acoustic wave.

A geometric interpretation of the Peano Axioms as a didactic strategy for teaching and learning natural numbers

Peano’s axioms are a set of propositions that allow a formal definition of the set of natural numbers. In these axioms, Peano rescues the essence of counting objects and materializes it by assuming the existence of a set, of a function in this set (which he calls successor) and an axiom of induction. In this work a geometrical representation and interpretation of each of these axioms is made, which allows understanding the independence and necessity of each one of them, which facilitates giving the conceptual rigor of them by making use of mathematical language. Finally, using Euclidean geometry, a geometric construction of a system of natural numbers on a straight line is made.

Computerized Generation of Endodontic Files by Reproducing the Flute Grinding Manufacturing Process.

Background: File fracture during root canal treatment in endodontics is a major concern for clinicians. The strength of the file is strongly dependent on its geometry, material, and working conditions; finite element simulations are used to understand these failure mechanisms. One limitation of the models used for these simulations is the approximate geometric representation typically obtained by rotating and scaling a specific cross-section shape along the file length. Given the influence of file geometry on file strength, a more realistic representation based on the manufacturing method is needed. Methods: A computerized method was developed to generate the file geometry by simulating the flute grinding manufacturing process. This method generates the 3D geometry of the file starting from a blank and reproducing the motions of the file and grinding wheel. Results: The cross-section of the resulting geometry does not involve simple rotation and scaling but changes from the shank to the tip. The tilt angle of the grinding wheel affects the final geometry, thus altering the convexity of the cross-section. Several other parameters, such as the pitch and the radius of the grinding disc tip, impact the final geometry. Conclusions: The proposed computational method allows for the generation of endodontic file geometries that match those produced via the actual flute grinding method. This tool may help researchers and tool designers in the preparation of finite element models to assess the strength of realistic files.

Simplicits: Mesh-Free, Geometry-Agnostic Elastic Simulation

The proliferation of 3D representations, from explicit meshes to implicit neural fields and more, motivates the need for simulators agnostic to representation. We present a data-, mesh-, and grid-free solution for elastic simulation for any object in any geometric representation undergoing large, nonlinear deformations. We note that every standard geometric representation can be reduced to an occupancy function queried at any point in space, and we define a simulator atop this common interface. For each object, we fit a small implicit neural network encoding spatially varying weights that act as a reduced deformation basis. These weights are trained to learn physically significant motions in the object via random perturbations. Our loss ensures we find a weight-space basis that best minimizes deformation energy by stochastically evaluating elastic energies through Monte Carlo sampling of the deformation volume. At runtime, we simulate in the reduced basis and sample the deformations back to the original domain. Our experiments demonstrate the versatility, accuracy, and speed of this approach on data including signed distance functions, point clouds, neural primitives, tomography scans, radiance fields, Gaussian splats, surface meshes, and volume meshes, as well as showing a variety of material energies, contact models, and time integration schemes.