Foundations of data science (Springfield, Mo.)最新文献

Topological data analysis (TDA) has had enormous success in science and engineering in the past decade. Persistent topological Laplacians (PTLs) overcome some limitations of persistent homology, a key technique in TDA, and provide substantial insight to the behavior of various geometric and topological objects. This work extends PTLs to directed flag complexes, which are an exciting generalization to flag complexes, also known as clique complexes, that arise naturally in many situations. We introduce the directed flag Laplacian and show that the proposed persistent directed flag Laplacian (PDFL) is a distinct way of analyzing these flag complexes. Example calculations are provided to demonstrate the potential of the proposed PDFL in real world applications.

Recently, various types of topological Laplacians have been studied from the perspective of data analysis. The spectral theory of these Laplacians has significantly extended the scope of algebraic topology and data analysis. Inspired by the theory of persistent Laplacians and cellular sheaves, this work develops the theory of persistent sheaf Laplacians for cellular sheaves and describes how to construct sheaves for a point cloud where each point is associated with a quantity that can be devised to embed physical properties. The spectra of persistent sheaf Laplacians encode both geometrical and non-geometrical information of the given point cloud. The theory of persistent sheaf Laplacians provides an elegant method for fusing different types of data and has significant potential for future development.

Human tissues are highly organized structures with specific collagen fiber arrangements varying from point to point. The effects of such heterogeneity play an important role for tissue function, and hence it is of critical to discover and understand the distribution of such fiber orientations from experimental measurements, such as the digital image correlation data. To this end, we introduce the heterogeneous peridynamic neural operator (HeteroPNO) approach, for data-driven constitutive modeling of heterogeneous anisotropic materials. The goal is to learn both a nonlocal constitutive law together with the material microstructure, in the form of a heterogeneous fiber orientation field, from loading field-displacement field measurements. To this end, we propose a two-phase learning approach. Firstly, we learn a homogeneous constitutive law in the form of a neural network-based kernel function and a nonlocal bond force, to capture complex homogeneous material responses from data. Then, in the second phase we reinitialize the learnt bond force and the kernel function, and training them together with a fiber orientation field for each material point. Owing to the state-based peridynamic skeleton, our HeteroPNO-learned material models are objective and have the balance of linear and angular momentum guaranteed. Moreover, the effects from heterogeneity and nonlinear constitutive relationship are captured by the kernel function and the bond force respectively, enabling physical interpretability. As a result, our HeteroPNO architecture can learn a constitutive model for a biological tissue with anisotropic heterogeneous response undergoing large deformation regime. The anisotropy and heterogeneity of this tissue stems from collagen fibers with unknown natural orientation, resulting in a location-dependent anisotropy. To demonstrate the applicability of our approach, we apply the heterogeneous PNO in learning the material model and fiber orientation field from digital image correction (DIC) data containing the planar displacement field on the tissue and the reaction forces in a biaxial testing. We find the learnt fiber architecture consistent with observations from polarized spatial frequency domain imaging. Moreover, the framework is capable to provide displacement and stress field predictions for new and unseen loading instances.

In algebraic topology, the differential (i.e., boundary operator) typically satisfies $d^2=0$ . However, the generalized differential $d^N=0$ for an integer $N\ge 2$ has been studied in terms of Mayer homology on $N$ -chain complexes for more than eighty years. We introduce Mayer Laplacians on $N$ -chain complexes. We show that both Mayer homology and Mayer Laplacians offer considerable application potential, providing topological and geometric insights to spaces. We also introduce persistent Mayer homology and persistent Mayer Laplacians at various $N$ . The bottleneck distance and stability of persistence diagrams associated with Mayer homology are investigated. Our computational experiments indicate that the topological features offered by persistent Mayer homology and spectrum given by persistent Mayer Laplacians hold substantial promise for large, complex, and diverse data. We envision that the present work serves as an inaugural step towards integrating Mayer homology and Mayer Laplacians into the realm of topological data analysis.

ChatGPT represents a significant milestone in the field of artificial intelligence (AI), finding widespread applications across diverse domains. However, its effectiveness in mathematical contexts has been somewhat constrained by its susceptibility to conceptual errors. Concurrently, topological data analysis (TDA), a relatively new discipline, has garnered substantial interest in recent years. Nonetheless, the advancement of TDA is impeded by the limited understanding of computational algorithms and coding proficiency among theoreticians. This work endeavors to bridge the gap between theoretical topological concepts and their practical implementation in computational topology through the utilization of ChatGPT. We showcase how a pure theoretician, devoid of computational experience and coding skills, can effectively transform mathematical formulations and concepts into functional codes for computational topology with the assistance of ChatGPT. Our strategy outlines a productive process wherein a mathematician trains ChatGPT on pure mathematical concepts, steers ChatGPT towards generating computational topology codes, and subsequently validates the generated codes using established examples. Our specific case studies encompass the computation of Betti numbers, Laplacian matrices, and Dirac matrices for simplicial complexes, as well as the persistence of various homologies and Laplacians. Furthermore, we explore the application of ChatGPT in computing recently developed topological theories for hypergraphs and digraphs, as well as the persistent harmonic space, which has not been computed in the literature, to the best of our knowledge. This work serves as an initial step towards effectively transforming pure mathematical theories into practical computational tools, with the ultimate goal of enabling real applications across diverse fields.

This work introduces the development of path Dirac and hypergraph Dirac operators, along with an exploration of their persistence. These operators excel in distinguishing between harmonic and non-harmonic spectra, offering valuable insights into the subcomplexes within these structures. The paper showcases the functionality of these operators through a series of examples in various contexts. An essential facet of this research involves examining the operators' sensitivity to filtration, emphasizing their capacity to adapt to topological changes. The paper also explores a significant application of persistent path Dirac and persistent hypergraph Dirac in molecular science, specifically in analyzing molecular structures. The study introduces strict preorders derived from molecular structures, which generate graphs and digraphs with intricate path structures. The depth of information within these path complexes reflects the complexity of different preorder classes influenced by molecular structures. This characteristic underscores the effectiveness of these tools in the realm of topological data analysis.

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.