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arXiv 2025年

作者： Kim, Jaeuk Torquato, Salvatore Princeton Materials Institute Department of Physics Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Materials Institute Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered stealthy hyperuniform (SHU) packings are an emerging class of exotic amorphous two-phase materials endowed with novel optical, transport, and mechanical properties. Such packings of identical spheres have been created from SHU ground-state point patterns via a modified collective-coordinate optimization scheme that includes a soft-core repulsion, besides the standard "stealthy" pair potential. To explore maximal ranges of the packing fraction , we investigate the distributions of minimum pair distances as well as nearest-neighbor distances of ensembles of SHU point patterns without and with soft-core repulsions in the first three space dimensions as a function of the stealthiness parameter χ and number of particles N within a hypercubic simulation box under periodic boundary conditions. Within the disordered regime (χ max(χ, d), decrease to zero on average as N increases if there are no soft-core repulsions. By contrast, the inclusion of soft-core repulsions results in very large max(χ, d) independent of N, reaching up to max(χ, d) = 1.0, 0.86, 0.63 in the zero-χ limit and decreasing to max(χ, d) = 1.0, 0.67, 0.47 at χ = 0.45 for d = 1, 2, 3, respectively. We obtain explicit formulas for max(χ, d) as functions of χ and N for a given value of d in both cases with and without soft-core repulsions. In two and three dimensions, our soft-core SHU ground-state packings for small χ become configurationally very close to the corresponding jammed hard-particle packings created by fast compression algorithms, as measured by their pair statistics. As χ increases beyond 0.20, the packings form fewer contacts and linear polymer-like chains as χ tends to 1/2. The resulting structure factors S(k) and pair correlation functions g2(r) reveal that soft-core repulsions significantly alter the short- and intermediate-range correlations in the SHU ground states. We show that the degree of large-scale order of the soft-core SHU ground states increases as χ increases from 0 to

关键词： Digital arithmetic

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On the importance of assessing topological convergence in Bayesian phylogenetic inference

arXiv

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arXiv 2024年

作者： Brusselmans, Marius Carvalho, Luiz Max Hong, Samuel L. Gao, Jiansi Matsen, Frederick A. Rambaut, Andrew Lemey, Philippe Suchard, Marc A. Dudas, Gytis Baele, Guy Department of Microbiology Immunology and Transplantation Rega Institute KU Leuven Leuven Belgium School of Applied Mathematics Getulio Vargas Foundation Praia de Botafogo 190 Rio de Janeiro22250-900 Brazil Computational Biology Program Fred Hutchinson Cancer Center SeattleWA98109 United States Howard Hughes Medical Institute Computational Biology Program Fred Hutchinson Cancer Research Center SeattleWA United States Department of Genome Sciences University of Washington SeattleWA United States Department of Statistics University of Washington SeattleWA United States Institute of Ecology and Evolution University of Edinburgh EdinburghEH9 3FL United Kingdom Department of Biostatistics Fielding School of Public Health University of California Los AngelesCA90095 United States Institute of Biotechnology Life Sciences Center Vilnius University Vilnius Lithuania

Modern phylogenetics research is often performed within a Bayesian framework, using sampling algorithms such as Markov chain Monte Carlo (MCMC) to approximate the posterior distribution. These algorithms require careful evaluation of the quality of the generated samples. Within the field of phylogenetics, one frequently adopted diagnostic approach is to evaluate the effective sample size (ESS) and to investigate trace graphs of the sampled parameters. A major limitation of these approaches is that they are developed for continuous parameters and therefore incompatible with a crucial parameter in these inferences: the tree topology. Several recent advancements have aimed at extending these diagnostics to topological space. In this reflection paper, we present two case studies – one on Ebola virus and one on HIV – illustrating how these topological diagnostics can contain information not found in standard diagnostics, and how decisions regarding which of these diagnostics to compute can impact inferences regarding MCMC convergence and mixing. Our results show the importance of running multiple replicate analyses and of carefully assessing topological convergence using the output of these replicate analyses. To this end, we illustrate different ways of assessing and visualizing the topological convergence of these replicates. Given the major importance of detecting convergence and mixing issues in Bayesian phylogenetic analyses, the lack of a unified approach to this problem warrants further action, especially now that additional tools are becoming available to researchers. © 2024, CC BY-SA.

关键词： Phylogenetic

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Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: I the Ergodic case.

arXiv

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arXiv 2019年

作者： Carmona, René Laurière, Mathieu Program in Applied and Computational Mathematics and Orfe

We propose two algorithms for the solution of the optimal control of ergodic McKean-Vlasov dynamics. Both algorithms are based on approximations of the theoretical solutions by neural networks, the latter being characterized by their architecture and a set of parameters. This allows the use of modern machine learning tools, and efficient implementations of stochastic gradient *** first algorithm is based on the idiosyncrasies of the ergodic optimal control problem. We provide a mathematical proof of the convergence of the approximation scheme, and we analyze rigorously the approximation by controlling the different sources of error. The second method is an adaptation of the deep Galerkin method to the system of partial differential equations issued from the optimality condition. We demonstrate the efficiency of these algorithms on several numerical examples, some of them being chosen to show that our algorithms succeed where existing ones failed. We also argue that both methods can easily be applied to problems in dimensions larger than what can be found in the existing literature. Finally, we illustrate the fact that, although the first algorithm is specifically designed for mean field control problems, the second one is more general and can also be applied to the partial differential equation systems arising in the theory of mean field games. Copyright © 2019, The Authors. All rights reserved.

关键词： Machine learning

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The Role of Heterogeneity in Autonomous Perimeter Defense Problems

arXiv

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arXiv 2022年

作者： Adler, Aviv Mickelin, Oscar Ramachandran, K. Sukhatme, Gaurav S. Karaman, Sertac Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology CambridgeMA United States Program in Applied and Computational Mathematics Princeton University Princeton NJ08544 United States Department of Computer Science University of Southern California Los AngelesCA90089 United States Department of Aeronautics and Astronautics Massachusetts Institute of Technology CambridgeMA02139 United States

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant. Copyright © 2022, The Authors. All rights reserved.

关键词： Dynamic programming

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Thermal Conductivity of Water at Extreme Conditions

arXiv

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arXiv 2023年

作者： Zhang, Cunzhi Puligheddu, Marcello Zhang, Linfeng Car, Roberto Galli, Giulia Pritzker School of Molecular Engineering University of Chicago ChicagoIL60637 United States Materials Science Division Center for Molecular Engineering Argonne National Laboratory LemontIL60439 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Chemistry University of Chicago ChicagoIL60637 United States

Measuring the thermal conductivity (κ) of water at extreme conditions is a challenging task and few experimental data are available. We predict κ for temperatures and pressures relevant to the conditions of the Earth mantle, between 1,000 and 2,000 K and up to 22 GPa. We employ close to equilibrium molecular dynamics simulations and a deep neural network potential fitted to density functional theory data. We then interpret our results by computing the equation of state of water on a fine grid of points and using a simple model for κ. We find that the thermal conductivity is weakly dependent on temperature and monotonically increases with pressure with an approximate square-root behavior. In addition we show how the increase of κ at high pressure, relative to ambient conditions, is related to the corresponding increase in the sound velocity. Although the relationships between the thermal conductivity, pressure and sound velocity established here are not rigorous, they are sufficiently accurate to allow for a robust estimate of the thermal conductivity of water in a broad range of temperatures and pressures, where experiments are still difficult to perform. Copyright © 2023, The Authors. All rights reserved.

关键词： Thermal conductivity

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Subdiffusive wave transport and weak localization transition in three-dimensional stealthy hyperuniform disordered systems

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Physical Review B 2022年第6期105卷 064204-064204页

作者： F. Sgrignuoli S. Torquato L. Dal Negro Department of Electrical and Computer Engineering & Photonics Center Boston University 8 Saint Mary's Street Boston Massachusetts 02215 USA Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA Division of Material Science and Engineering Boston University 15 Saint Mary's Street Brookline Massachusetts 02446 USA Department of Physics Boston University 590 Commonwealth Avenue Boston Massachusetts 02215 USA

The purpose of this work is to understand the fundamental connection between structural correlations and light localization in three-dimensional (3D) open scattering systems of finite size. We numerically investigate the transport of vector electromagnetic waves scattered by resonant electric dipoles spatially arranged in 3D space by stealthy hyperuniform disordered point patterns. Three-dimensional stealthy hyperuniform disordered systems are engineered with different structural correlation properties determined by their degree of stealthiness χ. Such fine control of exotic states of amorphous matter enables the systematic design of optical media that interpolate in a tunable fashion between uncorrelated random structures and crystalline materials. By solving the electromagnetic multiple scattering problem using Green's matrix spectral method, we establish a transport phase diagram that demonstrates a distinctive transition from a diffusive to a weak localization regime beyond a critical scattering density that depends on χ. The transition is characterized by studying the Thouless number and the spectral statistics of the scattering resonances. In particular, by tuning the χ parameter, we demonstrate large spectral gaps and suppressed subradiant proximity resonances, facilitating light localization. Moreover, consistently with previous studies, our results show a region of the transport phase diagram where the investigated scattering systems become transparent. Our work provides a systematic description of the transport and weak localization properties of light in stealthy hyperuniform structures and motivates the engineering of photonic systems with enhanced light-matter interactions for applications to both classical and quantum devices.

关键词： Anderson localization Light-matter interaction Mesoscopics Photonics Scattering theory Weak localization Metamaterials

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Beyond trees: classification with sparse pairwise dependencies

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2020年第1期21卷 7728-7760页

作者： Yaniv Tenzer Amit Moscovich Mary Frances Dorn Boaz Nadler Clifford Spiegelman Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot Israel Program in Applied and Computational Mathematics Princeton University Princeton NJ Los Alamos National Laboratory Los Alamos NM Department of Statistics Texas A&M University College Station TX

Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attaining low classification errors. In this setting, the optimal classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by a sparse linear combination of the univariate and bivariate log-transformed densities. Our proposed approach is semi-parametric in nature: we non-parametrically estimate the univariate and bivariate densities, remove pairs of variables that are nearly independent using the Hilbert-Schmidt independence criterion, and finally construct a linear SVM using the retained log-transformed densities. We demonstrate on synthetic and real data sets, that our classifier, named SLB (sparse log-bivariate density), is competitive with other popular classification methods.

关键词： binary classification graphical model Bayesian network sparsity semi-parametric

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Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions

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Physical Review X 2021年第2期11卷 021028-021028页

作者： Salvatore Torquato Jaeuk Kim Michael A. Klatt Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik FAU Erlangen-Nürnberg Staudtstr. 7 91058 Erlangen Germany

The local number variance σ2(R) associated with a spherical sampling window of radius R enables a classification of many-particle systems in d-dimensional Euclidean space Rd according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness γ1(R), excess kurtosis γ2(R), and the corresponding probability distribution function P[N(R)] of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for γ1(R) and γ2(R) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on γ1(R), γ2(R), and P[N(R)] for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for γ1(R), γ2(R), and P[N(R)] are generated for each model. We also ascertain the proximity of P[N(R)] to the normal distribution via a novel Gaussian “distance” metric l2(R). Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes in two or higher dimensions such that γ1(R)∼l2(R)∼R−(d+1)/2 and γ2(R)∼R−(d+1) for large R. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the “antihyperuniform” model studied here. We prove that one-dimensional hyperuniform systems of class I or any d-dimensional lattice cannot obey a CLT. Remarkably, we discover a type of universality in that, for all of ou

关键词： Exotic phases of matter Fluctuations & noise Nonequilibrium statistical mechanics Surface & interfacial phenomena

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Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks

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Physical Review Letters 2021年第3期127卷 037401-037401页

作者： Michael A. Klatt Paul J. Steinhardt Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik University of Erlangen-Nürnberg Staudtstr. 7 91058 Erlangen Germany Department of Chemistry Princeton Institute for the Science and Technology of Materialsand Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast α for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity S(α)≡dΔ(α)/dα, the first derivative of the optimal gap-to-midgap ratio Δ(α). In particular, for all our crystal networks, S(α) takes a universal form that is well approximated by the analytic formula for a 1D quarter-wave stack, SQWS(α). Even more surprisingly, the values of S(α) for our disordered networks converge to SQWS(α) for sufficiently large α. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.

关键词： Photonic crystals Diamond structure Networks & random structures Random & disordered media Band structure methods

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Hyperuniformity of maximally random jammed packings of hyperspheres across spatial dimensions

arXiv

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arXiv 2023年

作者： Maher, Charles Emmett Jiao, Yang Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Materials Science and Engineering Arizona State University TempeAZ85287 United States Department of Physics Arizona State University TempeAZ85287 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The maximally random jammed (MRJ) state is the most random (i.e., disordered) configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength density fluctuations are anomalously suppressed compared to typical disordered systems, i.e., their structure factors S(k) tend to zero as the wave number |k| tends to zero. Here, we show that generating high-quality strictly jammed states for Euclidean space dimensions d = 3, 4, and 5 is of paramount importance in ensuring hyperuniformity and extracting precise values of the hyperuniformity exponent α > 0 for MRJ states, defined by the power-law behavior of S(k) ∼ |k|α in the limit |k| → 0. Moreover, we show that for fixed d it is more difficult to ensure jamming as the particle number N increases, which results in packings that are nonhyperuniform. Free-volume theory arguments suggest that the ideal MRJ state does not contain rattlers, which act as defects in numerically generated packings. As d increases, we find that the fraction of rattlers decreases substantially. Our analysis of the largest truly jammed packings suggests that the ideal MRJ packings for all dimensions d ≥ 3 are hyperuniform with α = d − 2, implying the packings become more hyperuniform as d increases. The differences in α between MRJ packings and recently proposed Manna-class random close packed (RCP) states, which were reported to have α = 0.25 in d = 3 and be nonhyperuniform (α = 0) for d = 4 and d = 5, demonstrate the vivid distinctions between the large-scale structure of RCP and MRJ states in these dimensions. Our paper clarifies the importance of the link between true jamming and hyperuniformity and motivates the development of an algorithm to produce rattler-free three-dimensional MRJ packings. © 2023, CC BY.

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