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Hyperuniformity of generalized random organization models

Physical Review E 2019年第2期99卷 022115-022115页

作者： Zheng Ma Salvatore Torquato 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 08544 USA

Studies of random organization models of monodisperse (i.e., identical) spherical particles have shown that a hyperuniform state is achievable when the system goes through an absorbing phase transition to a critical state. Here we investigate to what extent hyperuniformity is preserved when the model is generalized to particles with a size distribution and/or nonspherical shapes. We begin by examining binary disks in two dimensions and demonstrate that their critical states are hyperuniform as two-phase media, but not hyperuniform nor multihyperuniform as point patterns formed by the particle centroids. We further confirm the generality of our findings by studying particles with a continuous size distribution. Finally, to study the effect of rotational degrees of freedom, we extend our model to noncircular particles, namely, hard rectangles with various aspect ratios, including the hard-needle limit. Although these systems exhibit only short-range orientational order, hyperuniformity is still preserved. Our analysis reveals that the redistribution of the “mass” of the particles rather than the particle centroids is central to this dynamical process. The consideration of the “active volume fraction” of generalized random organization models may help to resolve which universality class they belong to and hence may lead to a deeper theoretical understanding of absorbing-state models. Our results suggest that general particle systems subject to random organization can be a robust way to fabricate a wide class of hyperuniform states of matter by tuning the structures via different particle-size and -shape distributions. This in turn potentially enables the creation of multifunctional hyperuniform materials with desirable optical, transport, and mechanical properties.

关键词： Classical statistical mechanics

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Uncertainty quantification by random measures and fields

arXiv

引用

arXiv 2020年

作者： Bastian, Caleb Deen Rabitz, Herschel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Massive Dynamics PrincetonNJ08542 United States Department of Chemistry Princeton University PrincetonNJ08544 United States

We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk functionals of input-output maps. These are the ANOVA decomposition of the intensity measure and the decomposition of the random measure variance, each into subspaces. Orthogonal random measures furnish sensitivity distributions. We show that the random counting measure may be used to construct positive random fields, which admit decompositions of covariance and sensitivity indices and may be used to represent interacting particle systems. The first and second order global sensitivity analyses conveyed through random counting measures elucidate and integrate different notions of uncertainty quantification, and the global sensitivity analysis of random fields conveys the proportionate functional contributions to covariance. This framework complements others when used in conjunction with for instance algorithmic uncertainty and model selection uncertainty *** Codes 60G57, 60G60, 60G55 © 2020, CC-BY-NC-ND.

关键词： Uncertainty analysis

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Towards Implementation of the Pressure-Regulated, Feedback-Modulated Model of Star Formation in Cosmological Simulations: Methods and Application to TNG

arXiv

引用

arXiv 2024年

作者： Hassan, Sultan Ostriker, Eve C. Kim, Chang-Goo Bryan, Greg L. Burger, Jan D. Fielding, Drummond B. Forbes, John C. Genel, Shy Hernquist, Lars Jeffreson, Sarah M.R. Motwani, Bhawna Smith, Matthew C. Somerville, Rachel S. Steinwandel, Ulrich P. Teyssier, Romain Center for Cosmology and Particle Physics Department of Physics New York University 726 Broadway New YorkNY10003 United States Center for Computational Astrophysics Flatiron Institute 162 5th Ave New YorkNY10010 United States Department of Physics & Astronomy University of the Western Cape Cape Town7535 South Africa Department of Astrophysical Sciences Princeton University PrincetonNJ08544 United States Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States Department of Astronomy Columbia University 550 W 120th Street New YorkNY10027 United States Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 1 GarchingD-85748 Germany Department of Astronomy Cornell University IthacaNY14853 United States School of Physical and Chemical Sciences-Te Kura Matū University of Canterbury Private Bag 4800 Christchurch8140 New Zealand Columbia Astrophysics Laboratory Columbia University 550 West 120th Street New YorkNY10027 United States Center for Astrophysics Harvard & Smithsonian 60 Garden Street CambridgeMA United States Program in Applied and Computational Mathematics Princeton University Fine Hall Washington Road PrincetonNJ08544-1000 United States

Traditional star formation subgrid models implemented in cosmological galaxy formation simulations, such as that of Springel & Hernquist (2003, hereafter SH03), employ adjustable parameters to satisfy constraints measured in the local Universe. In recent years, however, theory and spatially-resolved simulations of the turbulent, multiphase, star-forming ISM have begun to produce new first-principles models, which when fully developed can replace traditional subgrid prescriptions. This approach has advantages of being physically motivated and predictive rather than empirically tuned, and allowing for varying environmental conditions rather than being tied to local Universe conditions. As a prototype of this new approach, by combining calibrations from the TIGRESS numerical framework with the Pressure-Regulated Feedback-Modulated (PRFM) theory, simple formulae can be obtained for both the gas depletion time and an effective equation of state. Considering galaxies in TNG50, we compare the "native" simulation outputs with post-processed predictions from PRFM. At TNG50 resolution, the total midplane pressure is nearly equal to the total ISM weight, indicating that galaxies in TNG50 are close to satisfying vertical equilibrium. The measured gas scale height is also close to theoretical equilibrium predictions. The slopes of the effective equations of states are similar, but with effective velocity dispersion normalization from SH03 slightly larger than that from current TIGRESS simulations. Because of this and the decrease in PRFM feedback yield at high pressure, the PRFM model predicts shorter gas depletion times than the SH03 model at high densities and redshift. Our results represent a first step towards implementing new, numerically calibrated subgrid algorithms in cosmological galaxy formation simulations. Copyright © 2024, The Authors. All rights reserved.

关键词： Cosmology

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Exponential acceleration of macroscopic quantum tunneling in a Floquet Ising model

arXiv

引用

arXiv 2023年

作者： Grattan, George Barton, Brandon A. Feeney, Sean Mossi, Gianni Patnaik, Pratik Sagal, Jacob C. Carr, Lincoln D. Oganesyan, Vadim Kapit, Eliot Quantum Engineering Program Colorado School of Mines 1523 Illinois St CO Golden80401 United States Department of Computer Science Colorado School of Mines 1500 Illinois St CO Golden80401 United States Department of Applied Mathematics and Statistics Colorado School of Mines 1500 Illinois St CO Golden80401 United States Department of Physics Colorado School of Mines 1523 Illinois St CO Golden80401 United States KBR Inc. 601 Jefferson St. HoustonTX77002 United States NASA Ames Research Center Moffett FieldCA94035 United States Department of Physics and Astronomy College of Staten Island CUNY Staten IslandNY10314 United States Physics program and Initiative for the Theoretical Sciences The Graduate Center CUNY New YorkNY10016 United States Center for Computational Quantum Physics Flatiron Institute 162 5th Avenue New YorkNY10010 United States

The exponential suppression of macroscopic quantum tunneling (MQT) in the number of elements to be reconfigured is an essential element of broken symmetry phases. Slow MQT is also a core bottleneck in quantum algorithms, such as traversing an energy landscape in optimization, and adiabatic state preparation more generally. In this work, we demonstrate the possibility to accelerate MQT through Floquet engineering with the application of a uniform, high frequency transverse drive field. Using the ferromagnetic phase of the transverse field Ising model in one and two dimensions as a prototypical example, we identify three qualitatively distinct regimes as a function of drive strength: (i) for weak drive, the system exhibits exponentially slow MQT alongside robust magnetic order, as expected;(ii) at intermediate drive strength, we find polynomial decay of rates alongside vanishing magnetic order consistent with critical or paramagnetic state;(iii) at very strong drive strengths both the tunnelling rate and time-averaged magnetic order remain finite with increasing system size. We support these claims with extensive full wavefunction and matrix-product state numerical simulations, and theoretical analysis. An experimental test of these results presents a technologically important and novel scientific question accessible on NISQ-era quantum computers. © 2023, CC BY.

关键词： Ising model

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mathematics for cryo-electron microscopy

Mathematics for cryo-electron microscopy

引用

2018 International Congress of Mathematicians, ICM 2018

作者： Singer, Amit Department of Mathematics Program in Applied and Computational Mathematic Princeton University United States

ISBN: (纸本)9789813272934

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. © ICM *** rights reserved.

关键词： X ray crystallography

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A Comparative Analysis of Optimization and Generalization Properties of Two-layer Neural Network and Random Feature Models Under Gradient Descent Dynamics

arXiv

引用

arXiv 2019年

作者： Weinan, E. Ma, Chao Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert *** Codes 41A99, 49M99 Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

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A Priori Estimates of the Population Risk for Residual Networks

arXiv

引用

arXiv 2019年

作者： Weinan, E. Ma, Chao Wang, Qingcan Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip connections and the nonlinearities differently so that paths with more nonlinearities are regularized by larger weights. The error estimates are a priori in the sense that the estimates depend only on the target function, not on the parameters obtained in the training process. The estimates are optimal, in a high dimensional setting, in the sense that both the bound for the approximation and estimation errors are comparable to the Monte Carlo error rates. A crucial step in the proof is to establish an optimal bound for the Rademacher complexity of the residual networks. Comparisons are made with existing norm-based generalization error bounds. Copyright © 2019, The Authors. All rights reserved.

关键词： Error analysis

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Machine learning from a continuous viewpoint

arXiv

引用

arXiv 2019年

作者： Weinan, E. Ma, Chao Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, very much in the spirit of classical numerical analysis and statistical physics. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the shallow neural network model and the residual neural network model, can all be recovered as particular discretizations of different continuous formulations. We also present examples of new models, such as the ow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework. Copyright © 2019, The Authors. All rights reserved.

关键词： Machine learning

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Analysis of the gradient descent algorithm for a deep neural network model with skip-connections

arXiv

引用

arXiv 2019年

作者： Weinan, E. Ma, Chao Wang, Qingcan Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

The behavior of the gradient descent (GD) algorithm is analyzed for a deep neural network model with skip-connections. It is proved that in the over-parametrized regime, for a suitable initialization, with high probability GD can find a global minimum exponentially fast. Generalization error estimates along the GD path are also established. As a consequence, it is shown that when the target function is in the reproducing kernel Hilbert space (RKHS) with a kernel defined by the initialization, there exist generalizable earlystopping solutions along the GD path. In addition, it is also shown that the GD path is uniformly close to the functions given by the related random feature model. Consequently, in this \implicit regularization" setting, the deep neural network model deteriorates to a random feature model. Our results hold for neural networks of any width larger than the input dimension. Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

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Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations

arXiv

引用

arXiv 2019年

作者： Jiequn, H.A.N. Chao, M.A. Zheng, M.A. Weinan, E. Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Department of Mathematics Purdue University Beijing Institute of Big Data Research

A new framework is introduced for constructing interpretable and truly reliable reduced models for multi-scale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first through an autoencoder to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active learning procedure is introduced to help ensure that the data set used is representative enough. The reduced system takes the form of the conventional moment systems and works regardless of the numerical discretization used. Numerical results are presented for the BGK model. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow. Copyright © 2019, The Authors. All rights reserved.

关键词： Hydrodynamics

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