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Learning dynamics of deep linear networks with multiple pathways 22

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Learning dynamics of deep linear networks with multiple path...

Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Jianghong Shi Eric Shea-Brown Michael A. Buice Department of Applied Mathematics University of Washington Seattle WA Department of Applied Mathematics University of Washington Seattle WA Allen Institute MindScope Program Seattle WA

ISBN: (纸本)9781713871088

Not only have deep networks become standard in machine learning, they are increasingly of interest in neuroscience as models of cortical computation that capture relationships between structural and functional properties. In addition they are a useful target of theoretical research into the properties of network computation. Deep networks typically have a serial or approximately serial organization across layers, and this is often mirrored in models that purport to represent computation in mammalian brains. There are, however, multiple examples of parallel pathways in mammalian brains. In some cases, such as the mouse, the entire visual system appears arranged in a largely parallel, rather than serial fashion. While these pathways may be formed by differing cost functions that drive different computations, here we present a new mathematical analysis of learning dynamics in networks that have parallel computational pathways driven by the same cost function. We use the approximation of deep linear networks with large hidden layer sizes to show that, as the depth of the parallel pathways increases, different features of the training set (defined by the singular values of the input-output correlation) will typically concentrate in one of the pathways. This result is derived analytically and demonstrated with numerical simulation with both linear and non-linear networks. Thus, rather than sharing stimulus and task features across multiple pathways, parallel network architectures learn to produce sharply diversified representations with specialized and specific pathways, a mechanism which may hold important consequences for codes in both biological and artificial systems.

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AN OPTIMAL SCHEDULED LEARNING RATE FOR A RANDOMIZED KACZMARZ ALGORITHM

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arXiv 2022年

作者： Marshall, Nicholas F. Mickelin, Oscar Department of Mathematics Oregon State University United States Program in Applied and Computational Mathematics Princeton University United States

We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving Ax ≈ b + Ε, where Ax = b is a consistent linear system and Ε has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases;in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-W function of an exponential. Copyright © 2022, The Authors. All rights reserved.

关键词： Linear systems

Ab initio multi-scale modeling of ferroelectrics: The case of PbTiO3

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arXiv 2022年

作者： Xie, Pinchen Chen, Yixiao Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States

We report an ab initio multi-scale study of lead titanate using the Deep Potential (DP) models, a family of machine learning-based atomistic models, trained on first-principles density functional theory data, to represent potential and polarization surfaces. Our approach includes anharmonic effects beyond the limitations of reduced models and of the linear approximation for the polarization. The calculated enthalpy, spontaneous polarization, specific heat and dielectric susceptibility agree well with experiments on single crystals. In addition, we study how the free energy depends on the polarization with enhanced sampling methods, further supporting the first-order and order-disorder character of the transition. The latter is evidenced by persistence of local dipoles above the transition temperature. The simulated free energy surface as a function of the global polarization leads to a Landau-Devonshire theory of the single domain crystal. © 2022, CC BY.

关键词： Free energy

Thermal disorder and phonon softening in the ferroelectric phase transition of lead titanate

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arXiv 2024年

作者： Xie, Pinchen Chen, Yixiao Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States AI for Science Institute Beijing China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Materials Institute Princeton University PrincetonNJ08544 United States

We report an extensive molecular dynamics study of ab-initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained on density functional theory data with modern machine learning techniques. Our simulations demonstrate that the transition has a strong order-disorder character, in agreement with diffraction experiments, and provide fresh insight into the approach to equilibrium upon crossing the transition temperature. We find that the macroscopic polarization appears/disappears due to dipolar switching at the nanometer scale. We also compute the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. For a long time, the almost ideal displacive character of the soft mode detected by Raman scattering in the paraelectric phase has been contrasted with the order-disorder character of the transition suggested by diffraction. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation due to the thermal disordering of the dipoles. The Debye relaxation feature is centered at zero frequency and appears near the transition temperature in both the ferroelectric and the paraelectric phases. © 2024, CC BY.

关键词： Ferroelectricity

Exploring 3D Reconstruction with Drone Images: Advances and Challenges in Urban Environments 50

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Exploring 3D Reconstruction with Drone Images: Advances and ...

50th Annual Conference of the IEEE Industrial Electronics Society, IECON 2024

作者： Miranda Baldivieso, Thiago João da Silva Batista, Taise Grazielle Chagas, Fabio Suim Rodrigues Velho, Luiz Carlos Pacheco Ferreira Rosa, Paulo Fernando Defense Engineering Graduate Program Military Institute of Engineering Rio de Janeiro Brazil State Institute of the Environment Inea RJ Rio de Janeiro Brazil Department of Science and Industry systems University of South-Eastern Norway Kongsberg Norway Department of Computer Graphics Institute for Pure and Applied Mathematics Rio de Janeiro Brazil Electrical Engineering Graduate Program Military Institute of Engineering Rio de Janeiro Brazil

ISBN: (纸本)9781665464543

This working-in-progress paper aims to present a three-dimensional reconstruction using aerial images in different environments. The experiments were conducted with aircraft in both external and internal settings, starting with image acquisition, followed by the application of specific photogrammetry software-both commercial and open-source-and concluding with a qualitative evaluation of the results. © 2024 IEEE.

关键词： Aerial photography

Deep potentials for materials science

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Materials Futures 2022年第2期1卷 89-115页

作者： Tongqi Wen Linfeng Zhang Han Wang Weinan E David J Srolovitz Department of Mechanical Engineering The University of Hong KongHong KongHong Kong Special Administrative Region of China DP Technology BeijingPeople’s Republic of China AI for Science Institute BeijingPeople’s Republic of China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China HEDPS CAPTCollege of EngineeringPeking UniversityBeijingPeople’s Republic of China School of Mathematical Sciences Peking UniversityBeijingPeople’s Republic of China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJUnited States of America International Digital Economy Academy(IDEA) ShenzhenPeople’s Republic of China

To fill the gap between accurate(and expensive)ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials,a new class of descriptions of atomic interactions has emerged and been widely applied;*** learning potentials(MLPs).One recently developed type of MLP is the deep potential(DP)*** this review,we provide an introduction to DP methods in computational materials *** theory underlying the DP method is presented along with a step-by-step introduction to their development and *** also review materials applications of DPs in a wide range of materials *** DP Library provides a platform for the development of DPs and a database of extant *** discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials.

关键词： deep potential atomistic simulation machine learning potential neural network

A DYNAMICAL SYSTEM PROOF OF NIVEN'S THEOREM AND ITS EXTENSIONS

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arXiv 2023年

作者： Panraksa, Chatchawan Samart, Detchat Sriwongsa, Songpon Applied Mathematics Program Mahidol University International College Nakhon Pathom73170 Thailand Department of Mathematics Faculty of Science Burapha University Chonburi20131 Thailand Department of Mathematics Faculty of Science King Mongkut's University of Technology Thonburi Bangkok10140 Thailand

Niven's theorem asserts that {cos(rπ) | r ∈ } ∩ = {0, ±1, ±1/2}. In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set {cos(rπ) | r ∈ } ∩ K, where K is an arbitrary number *** Codes 37C25, 11C08, 11R04, 33B10 © 2023, CC BY.

关键词： Dynamical systems

Statistical mechanics of thermostatically controlled multizone buildings

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Physical Review E 2023年第3期107卷 034140-034140页

作者： Lucas Fuentes Valenzuela Lindell Williams Michael Chertkov Department of Electrical Engineering Stanford University 350 Jane Stanford Way Stanford California 94305 USA Department of Physics Cornell University 109 Clark Hall Ithaca New York 14853 USA Program in Applied Mathematics & Department of Mathematics University of Arizona 617 N. Santa Rita Tucson Arizona 85721 USA

We study the collective phenomena and constraints associated with the aggregation of individual cooling units from a statistical mechanics perspective. These units are modeled as thermostatically controlled loads (TCLs) and represent zones in a large commercial or residential building. Their energy input is centralized and controlled by a collective unit—the air handling unit (AHU)—delivering cool air to all TCLs, thereby coupling them together. Aiming to identify representative qualitative features of the AHU-to-TCL coupling, we build a simple but realistic model and analyze it in two distinct regimes: the constant supply temperature (CST) and the constant power input (CPI) regimes. In both cases, we center our analysis on the relaxation dynamics of individual TCL temperatures to a statistical steady state. We observe that while the dynamics are relatively fast in the CST regime, resulting in all TCLs evolving around the control set point, the CPI regime reveals the emergence of a bimodal probability distribution and two, possibly strongly separated, timescales. We observe that the two modes in the CPI regime are associated with all TCLs being in the same low or high airflow states, with an occasional collective transition between the modes akin to Kramer's phenomenon in statistical physics. To the best of our knowledge, this phenomenon has been overlooked in building energy systems despite its direct operational implications. It highlights a trade-off between occupational comfort—related to zonal temperature variations—and energy consumption.

关键词： Electric power transmission Energy-efficient infrastructure Engineering Fluctuations & noise Irreversible processes Fokker–Planck equation Langevin algorithm Langevin equation Monte Carlo methods Physics & society

Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

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arXiv 2022年

作者： Shi, Yunpeng Li, Shaohan Maunu, Tyler Lerman, Gilad Program in Applied and Computational Mathematics Princeton University United States School of Mathematics University of Minnesota United States Department of Mathematics Brandeis University United States

We develop new statistics for robustly filtering corrupted keypoint matches in the structure from motion pipeline. The statistics are based on consistency constraints that arise within the clustered structure of the graph of keypoint matches. The statistics are designed to give smaller values to corrupted matches and than uncorrupted matches. These new statistics are combined with an iterative reweighting scheme to filter keypoints, which can then be fed into any standard structure from motion pipeline. This filtering method can be efficiently implemented and scaled to massive datasets as it only requires sparse matrix multiplication. We demonstrate the efficacy of this method on synthetic and real structure from motion datasets and show that it achieves state-of-the-art accuracy and speed in these tasks. Copyright © 2022, The Authors. All rights reserved.

关键词： Statistics