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Geometrical ambiguity of pair statistics: Point configurations

Physical Review E 2010年第1期81卷 011105-011105页

作者： Y. Jiao F. H. Stillinger S. Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA and Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

Point configurations have been widely used as model systems in condensed-matter physics, materials science, and biology. Statistical descriptors, such as the n-body distribution function gn, are usually employed to characterize point configurations, among which the most extensively used is the pair distribution function g2. An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in g2 is, in general, insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we present the idea of the distance space called the D space. The pair distances of a specific point configuration are then represented by a single point in the D space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function g2. Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations, i.e., with structural degeneracy. These conditions define a bounded region in the D space. By explicitly constructing a variety of degenerate point configurations using the D space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the D space, including the reconstruction of atomic structures from experimentally obtained g2 and a recently proposed “decorrelation” principle. The degenerate configurations have relevance to open questions involving the famous traveling salesman problem.

关键词： Inverse problems

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Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity

arXiv

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arXiv 2024年

作者： Zhang, Handi Liu, Langchen Lu, Lu Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Statistics and Data Science Yale University New HavenCT06511 United States Wu Tsai Institute Yale University New HavenCT06510 United States

By leveraging neural networks, the emerging field of scientific machine learning (SciML) offers novel approaches to address complex problems governed by partial differential equations (PDEs). In practical applications, challenges arise due to the distributed essence of data, concerns about data privacy, or the impracticality of transferring large volumes of data. Federated learning (FL), a decentralized framework that enables the collaborative training of a global model while preserving data privacy, offers a solution to the challenges posed by isolated data pools and sensitive data issues. Here, this paper explores the integration of FL and SciML to approximate complex functions and solve differential equations. We propose two novel models: federated physics-informed neural networks (FedPINN) and federated deep operator networks (FedDeepONet). We further introduce various data generation methods to control the degree of non-independent and identically distributed (non-iid) data and utilize the 1-Wasserstein distance to quantify data heterogeneity in function approximation and PDE learning. We systematically investigate the relationship between data heterogeneity and federated model performance. Additionally, we propose a measure of weight divergence and develop a theoretical framework to establish growth bounds for weight divergence in federated learning compared to traditional centralized learning. To demonstrate the effectiveness of our methods, we conducted 10 experiments, including 2 on function approximation, 5 PDE problems on FedPINN, and 3 PDE problems on FedDeepONet. These experiments demonstrate that proposed federated methods surpass the models trained only using local data and achieve competitive accuracy of centralized models trained using all data. © 2024, CC BY-NC-ND.

关键词： Federated learning

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Modeling heterogeneous materials via two-point correlation functions: Basic principles

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Physical Review E 2007年第3期76卷 031110-031110页

作者： Y. Jiao F. H. Stillinger S. Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton University Princeton New Jersey 08544 USA

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato [Phys. Rev. E 57, 495 (1998)] introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2(r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2(r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2(r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

关键词： RECONSTRUCTING RANDOM-MEDIA PROBABILITY FUNCTIONS NONLINEAR COMPOSITES POROUS-MEDIA BOUNDS MICROSTRUCTURES OPTIMIZATION SCATTERING

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Isotope effects on molecular structures and electronic properties of liquid water via deep potential molecular dynamics based on SCAN functional

arXiv

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arXiv 2020年

作者： Xu, Jianhang Zhang, Chunyi Zhang, Linfeng Chen, Mohan Santra, Biswajit Wu, Xifan Department of Physics Temple University PhiladelphiaPA19122 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States CAPT HEDPS College of Engineering Peking University Beijing100871 China Institute for Computational Molecular Science Temple University PhiladelphiaPA19122 United States

Feynman path-integral deep potential molecular dynamics (PI-DPMD) calculations have been employed to study both light (H2O) and heavy water (D2O) within the isothermal–isobaric ensemble. In particular, the deep neural network is trained based on ab initio data obtained from the strongly constrained and appropriately normed (SCAN) exchange-correlation functional. Because of the lighter mass of hydrogen than deuteron, the properties of light water is more influenced by nuclear quantum effect than those of heavy water. Clear isotope effects are observed and analyzed in terms of hydrogen-bond structure and electronic properties of water that are closely associated with experimental observables. The molecular structures of both liquid H2O and D2O agree well with the data extracted from scattering experiments. The delicate isotope effects on radial distribution functions and angular distribution functions are well reproduced as well. Our approach demonstrates that deep neural network combined with SCAN functional based ab initio molecular dynamics provides an accurate theoretical tool for modeling water and its isotope effects. Copyright © 2020, The Authors. All rights reserved.

关键词： Electronic properties

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An analysis of reconstruction noise from undersampled 4D flow MRI

arXiv

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arXiv 2022年

作者： Partin, Lauren Schiavazzi, Daniele E. Long, Carlos A. Sing Department Of Applied And Computational Mathematics And Statistics University Of Notre Dame Notre DameIN United States Institute For Mathematical And Computational Engineering Institute For Biological And Medical Engineering Pontificia Universidad Católica De Chile Santiago Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For The Discovery Of Structures In Complex Data Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For Cardiovascular Magnetic Resonance Chile

Novel Magnetic Resonance (MR) imaging modalities can quantify hemodynamics but require long acquisition times, precluding its widespread use for early diagnosis of cardiovascular disease. To reduce the acquisition times, reconstruction methods from undersampled measurements are routinely used, that leverage representations designed to increase image compressibility. Reconstructed anatomical and hemodynamic images may present visual artifacts. Although some of these artifact are essentially reconstruction errors, and thus a consequence of undersampling, others may be due to measurement noise or the random choice of the sampled frequencies. Said otherwise, a reconstructed image becomes a random variable, and both its bias and its covariance can lead to visual artifacts;the latter leads to spatial correlations that may be misconstrued for visual information. Although the nature of the former has been studied in the literature, the latter has not received as much attention. In this study, we investigate the theoretical properties of the random perturbations arising from the reconstruction process, and perform a number of numerical experiments on simulated and MR aortic flow. Our results show that the correlation length remains limited to two to three pixels when a Gaussian undersampling pattern is combined with recovery algorithms based on 1-norm minimization. However, the correlation length may increase significantly for other undersampling patterns, higher undersampling factors (i.e., 8× or 16× compression), and different reconstruction methods. © 2022, CC BY-NC-ND.

关键词： Magnetic resonance

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A novel quantum transition in a fully frustrated transverse Ising antiferromagnet

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Journal of Physics: Conference Series 2009年第1期143卷

作者： Chandra, Anjan Kumar Inoue, Jun-Ichi Chakrabarti, Bikas K Centre for Applied Mathematics and Computational Science Saha Institute of Nuclear Physics Kolkata-700064 1/AF Bidhannagar India Complex Systems Engineering Graduate School of Information Science and Technology Hokkaido University Kita-ku Sapporo 060-0814 N14-W9 Japan Centre for Applied Mathematics and Computational Science Theoretical Condensed Matter Physics Division Saha Institute of Nuclear Physics Kolkata-700064 1/AF Bidhannagar India

We consider a long-range Ising antiferromagnet (LRIAF) put in a transverse field. Applying quantum Monte Carlo method, we study the variation of order parameter (spin correlation in Trotter time direction), susceptibility and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We also discuss the phase diagram for the Sherrington-Kirkpatrick (SK) model with the same LRIAF bias, also in presence of a transverse field. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the system, an infinitesimal SK spin glass disorder is enough to induce a stable glass order in the antiferromagnet. This glass order eventually gets destroyed as the thermal or quantum fluctuations increased beyond their threshold values and the transition to para phase occurs. Indications of this novel phase transition are discussed. Because of the presence of full frustration, this surrogate property of the LRIAF for incubation of stable spin glass phase in it (induced by addition of a small disorder) should enable eventually the study of classical and quantum spin glass phases by using some perturbation theory with respect to the disorder. © 2009 IOP Publishing Ltd.

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Atomic photoabsorption process controlled by static and oscillating magnetic fields

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Physical Review A 2009年第4期80卷 043414-043414页

作者： Takeshi Shirahama Xiao-Min Tong Ken-ichi Hino Nobuyuki Toshima Institute of Materials Science University of Tsukuba 1-1-1 Tennodai Tsukuba Ibaraki 305-8573 Japan Center for Computational Sciences University of Tsukuba 1-1-1 Tennodai Tsukuba Ibaraki 305-8577 Japan Doctoral Program in Frontier Science Graduate School of Pure and Applied Sciences University of Tsukuba 1-1-1 Tennodai Tsukuba Ibaraki 305-8573 Japan

We present a theoretical method to study atomic photoabsorption processes in a temporally periodic external field. The time development of the atomic dipole moment is propagated by numerical time propagation, and the photoabsorption cross sections is obtained directly from the Fourier transform of the autocorrelation function without summing up over the Floquet-Fourier components. As an example, we study the photoabsorption spectra of hydrogen atoms exposed in a mixture of static and periodically oscillating magnetic fields for the transition from 1s→2p. The atomic energy structure and photoabsorption spectra show interesting dependence on the strength of the oscillating magnetic field. Tuning the combination of the magnetic field strengths, we can control the photoabsorption process. The proposed method is general and can be applied to processes in which the number of relevant high harmonic components of the Fourier expansion is so large that the conventional Floquet method is intractable.

关键词： Atoms

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A FLOW ARTIST FOR HIGH-DIMENSIONAL CELLULAR DATA

arXiv

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arXiv 2023年

作者： MacDonald, Kincaid Bhaskar, Dhananjay Thampakkul, Guy Nguyen, Nhi Zhang, Joia Perlmutter, Michael Adelstein, Ian Krishnaswamy, Smita Department of Mathematics Yale University United States Department of Genetics Yale School of Medicine United States Department of Computer Science Yale University United States Department of Mathematics Pomona College United States Applied Mathematics Program Yale University United States Department of Statistics University of Washington United States Department of Mathematics Boise State University United States Computational Biology and Bioinformatics Program Yale University United States

We consider the problem of embedding point cloud data sampled from an underlying manifold with an associated flow or velocity. Such data arises in many contexts where static snapshots of dynamic entities are measured, including in high-throughput biology such as single-cell transcriptomics. Existing embedding techniques either do not utilize velocity information or embed the coordinates and velocities independently, i.e., they either impose velocities on top of an existing point embedding or embed points within a prescribed vector field. Here we present FlowArtist, a neural network that embeds points while jointly learning a vector field around the points. The combination allows FlowArtist to better separate and visualize velocity-informed structures. Our results, on toy datasets and single-cell RNA velocity data, illustrate the value of utilizing coordinate and velocity information in tandem for embedding and visualizing high-dimensional data. © 2023, CC BY.

关键词： RNA

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PIRATENETS: PHYSICS-INFORMED DEEP LEARNING WITH RESIDUAL ADAPTIVE NETWORKS

arXiv

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arXiv 2024年

作者： Wang, Sifan Li, Bowen Perdikaris, Paris Chen, Yuhan Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Mathematics Duke University DurhamNC27708 United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania PhiladelphiaPA19104 United States Department of Electrical and Computer Engineering North Carolina State University RaleighNC27695 United States

While physics-informed neural networks (PINNs) have become a popular deep learning framework for tackling forward and inverse problems governed by partial differential equations (PDEs), their performance is known to degrade when larger and deeper neural network architectures are employed. Our study identifies that the root of this counter-intuitive behavior lies in the use of multi-layer perceptron (MLP) architectures with non-suitable initialization schemes, which result in poor trainablity for the network derivatives, and ultimately lead to an unstable minimization of the PDE residual loss. To address this, we introduce Physics-Informed Residual Adaptive Networks (PirateNets), a novel architecture that is designed to facilitate stable and efficient training of deep PINN models. PirateNets leverage a novel adaptive residual connection, which allows the networks to be initialized as shallow networks that progressively deepen during training. We also show that the proposed initialization scheme allows us to encode appropriate inductive biases corresponding to a given PDE system into the network architecture. We provide comprehensive empirical evidence showing that PirateNets are easier to optimize and can gain accuracy from considerably increased depth, ultimately achieving state-of-the-art results across various benchmarks. All code and data accompanying this manuscript will be made publicly available at https://***/PredictiveIntelligenceLab/jaxpi. © 2024, CC BY-NC-ND.

关键词： Inverse problems

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Retraction notice to "Hydrological and climate impacts on river characteristics of Pahang River Basin, Malaysia" [ Heliyon 9 2023 e21573]

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Heliyon 2025年第1期11卷 e41351页

作者： Mohd Khairul Amri Kamarudin Mohd Ekhwan Toriman Noorjima Abd Wahab Mohd Armi Abu Samah Khairul Nizam Abdul Maulud Firdaus Mohamad Hamzah Ahmad Shakir Mohd Saudi Sunardi Sunardi Faculty of Applied Social Science University Sultan Zainal Abidin Gong Badak Campus 21300 Terengganu Malaysia. East Coast Environmental Research Institute (ESERI) University Sultan Zainal Abidin Gong Badak Campus Nerus 21300 Kuala Malaysia. Faculty of Social Sciences and Humanities National University of Malaysia 43600 Bangi Selangor Malaysia. Kulliyyah of Science International Islamic University Malaysia 25200 Kuantan Pahang Malaysia. Earth Observation Center Institute of Climate Change National University of Malaysia 43600 Bangi Selangor Malaysia. Department of Mathematics Defense Base Center National Defence University of Malaysia Sungai Besi Camp 57000 Kuala Lumpur Malaysia. Department of Environmental Health Institute of Medical Science Technology Universiti Kuala Lumpur Kajang Selangor Malaysia. Graduate Program on Environmental Studies Postgraduate School Universitas Padjadjaran Jl. Dipati Ukur No. 35 Bandung 40132 Indonesia.

[This retracts the article DOI: 10.1016/***.2023.e21573.].

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