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Disordered Heterogeneous Universe: Galaxy Distribution and Clustering across Length Scales

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Physical Review X 2023年第1期13卷 011038-011038页

作者： Oliver H. E. Philcox Salvatore Torquato Department of Astrophysical Sciences Princeton University Princeton New Jersey 08540 USA School of Natural Sciences Institute for Advanced Study 1 Einstein Drive Princeton New Jersey 08540 USA Center for Theoretical Physics Department of Physics Columbia University New York New York 10027 USA Simons Foundation New York New York 10010 USA Department of Chemistry Department of Physics Princeton Materials Institute and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA

The studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the disordered distribution of particles and/or building blocks at microscales to predict physical properties of the medium at macroscales, whether it be a liquid, colloidal suspension, composite material, galaxy cluster, or entire Universe. The theory of disordered heterogeneous media provides an array of theoretical and computational techniques to characterize a wide class of complex material microstructures. In this work, we apply them to describe the disordered distributions of galaxies obtained from recent suites of dark matter simulations. We focus on the determination of lower-order correlation functions, void and particle nearest-neighbor functions, certain cluster statistics, pair-connectedness functions, percolation properties, and a scalar order metric to quantify the degree of order. Compared to analogous homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the void and particle nearest-neighbor functions, due to the presence of quasi-long-range correlations imprinted by early Universe physics, with a minimum particle separation far below the mean nearest-neighbor distance. On large scales, the system appears hyperuniform, as a result of primordial density fluctuations, while on the smallest scales, the system becomes almost antihyperuniform, as evidenced by its number variance. Additionally, via a finite-scaling analysis, we compute the percolation threshold of the galaxy catalogs, finding this to be significantly lower than for Poisson realizations (at reduced density ηc=0.25 in our fiducial analysis compared to ηc=0.34), with strong dependence on the mean density; this is consistent with the observation that the galaxy distribution contains voids of up to 50% larger radius. However, the two sets of simulations appear to share the same fractal dimension on scales much l

关键词： Large scale structure of the Universe

Learning A Coarse-to-Fine Diffusion Transformer for Image Restoration

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arXiv 2023年

作者： Wang, Liyan Yang, Qinyu Wang, Cong Wang, Wei Pan, Jinshan Su, Zhixun School of Mathematical Sciences Dalian University of Technology Dalian China The Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province The Liaoning Center for Applied Mathematics China The Department of Computing The Hong Kong Polytechnic University Hong Kong Dalian University of Technology Dalian China The School of Computer Science and Engineering Nanjing University of Science and Technology Nanjing China

Recent years have witnessed the remarkable performance of diffusion models in various vision tasks. However, for image restoration that aims to recover clear images with sharper details from given degraded observations, diffusion-based methods may fail to recover promising results due to inaccurate noise estimation. Moreover, simple constraining noises cannot effectively learn complex degradation information, which subsequently hinders the model capacity. To solve the above problems, we propose a coarse-to-fine diffusion Transformer (C2F-DFT) for image restoration. Specifically, our C2F-DFT contains diffusion self-attention (DFSA) and diffusion feed-forward network (DFN) within a new coarse-to-fine training scheme. The DFSA and DFN respectively capture the long-range diffusion dependencies and learn hierarchy diffusion representation to facilitate better restoration. In the coarse training stage, our C2FDFT estimates noises and then generates the final clean image by a sampling algorithm. To further improve the restoration quality, we propose a simple yet effective fine training scheme. It first exploits the coarse-trained diffusion model with fixed steps to generate restoration results, which then would be constrained with corresponding ground-truth ones to optimize the models to remedy the unsatisfactory results affected by inaccurate noise estimation. Extensive experiments show that C2F-DFT significantly outperforms diffusion-based restoration method IR-SDE and achieves competitive performance compared with Transformer-based state-of-the-art methods on 3 tasks, including image deraining, image deblurring, and real image denoising. Code is available at https://***/wlydlut/C2F-DFT. Copyright © 2023, The Authors. All rights reserved.

关键词： Image reconstruction

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium

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Physical Review E 2022年第4期106卷 044122-044122页

作者： Salvatore Torquato Haina Wang Department of Chemistry Department of Physics Princeton Institute of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study 1 Einstein Drive Princeton New Jersey 08540 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g2(r) or corresponding structure factor S(k) in d-dimensional Euclidean space Rd is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)] conjectured that any realizable g2(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g2(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform “cloaked” uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accur

关键词： Nonequilibrium statistical mechanics Optimization problems Structural properties Complex materials

Designer Pair Statistics of Disordered Many-Particle Systems with Novel Properties

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arXiv 2023年

作者： Wang, Haina Torquato, Salvatore 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 School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

Knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing data base of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperature that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics. © 2023, CC BY.

关键词： Statistics

Compatible L2 norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection

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arXiv 2022年

作者： Yang, Yin Wang, Jindi Chen, Yanping Liao, Hong-Lin School of Mathematics and Computational Science Xiangtan University Hunan National Applied Mathematics Center Hunan Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences South China Normal University Guangdong Guangzhou510631 China Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China MIIT Nanjing211106 China

MSC Codes 35Q99, 65M06, 65M12, 74A50The convergence of variable-step L1 scheme is studied for the time-fractional molecular beam epitaxy (MBE) model with slope selection. A novel asymptotically compatible L2 norm error estimate of the variable-step L1 scheme is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter Ε → 0+. To the best of our knowledge, it is the first time to establish such error estimate for nonlinear subdiffusion problems. The asymptotically compatible convergence means that the error estimate is compatible with that of backward Euler scheme for the classical MBE model as the fractional order α → 1-. Just as the backward Euler scheme can maintain the physical properties of the MBE equation, the variable-step L1 scheme can also preserve the corresponding properties of the time-fractional MBE model, including the volume conservation, variational energy dissipation law and L2 norm boundedness. Numerical experiments are presented to support our theoretical results. © 2022, CC BY-NC-ND.

关键词： Energy dissipation

From Kernels to Features: A Multi-Scale Adaptive Theory of Feature Learning

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arXiv 2025年

作者： Rubin, Noa Fischer, Kirsten Lindner, Javed Dahmen, David Seroussi, Inbar Ringel, Zohar Krämer, Michael Helias, Moritz The Racah Institute of Physics The Hebrew University of Jerusalem Jerusalem Israel Computational and Systems Neuroscience Jülich Research Centre Jülich Germany RWTH Aachen University Aachen Germany Department of Physics RWTH Aachen University Aachen Germany Institute for Theoretical Particle Physics and Cosmology RWTH Aachen University Aachen Germany Department of Applied Mathematics School of Mathematical Sciences Tel-Aviv University Tel-Aviv Israel

Theoretically describing feature learning in neural networks is crucial for understanding their expressive power and inductive biases, motivating various approaches. Some approaches describe network behavior after training through a simple change in kernel scale from initialization, resulting in a generalization power comparable to a Gaussian process. Conversely, in other approaches training results in the adaptation of the kernel to the data, involving complex directional changes to the kernel. While these approaches capture different facets of network behavior, their relationship and respective strengths across scaling regimes remains an open question. This work presents a theoretical framework of multi-scale adaptive feature learning bridging these approaches. Using methods from statistical mechanics, we derive analytical expressions for network output statistics which are valid across scaling regimes and in the continuum between them. A systematic expansion of the network’s probability distribution reveals that mean-field scaling requires only a saddle-point approximation, while standard scaling necessitates additional correction terms. Remarkably, we find across regimes that kernel adaptation can be reduced to an effective kernel rescaling when predicting the mean network output of a linear network. However, even in this case, the multi-scale adaptive approach captures directional feature learning effects, providing richer insights than what could be recovered from a rescaling of the kernel alone. © 2025, CC BY-NC-SA.

关键词： Contrastive Learning

Natural parameter conditions for singular perturbations of chemical and biochemical reaction networks

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arXiv 2022年

作者： Eilertsen, Justin Schnell, Santiago Walcher, Sebastian Mathematical Reviews American Mathematical Society 416 4th Street Ann ArborMI48103 United States Department of Biological Sciences Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Mathematik A RWTH AachenD-52056 Germany

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis–Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be *** Codes 92C45, 34D15, 80A30, 13P10 © 2022, CC BY.

关键词： Lyapunov functions

Predicting rare events using neural networks and short-trajectory data

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arXiv 2022年

作者： Strahan, John Finkel, Justin Dinner, Aaron R. Weare, Jonathan Department of Chemistry and James Franck Institute University of Chicago ChicagoIL60637 United States Committee on Computational and Applied Mathematics The University of Chicago ChicagoIL60637 United States Department of Chemistry James Franck Institute and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Courant Institute of Mathematical Sciences New York University New York10012 United States

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations). Here, we develop an approach to solve Feynman-Kac equations by training neural networks on short-trajectory data. Our approach is based on a Markov approximation but otherwise avoids assumptions about the underlying model and dynamics. This makes it applicable to treating complex computational models and observational data. We illustrate the advantages of our method using a low-dimensional model that facilitates visualization, and this analysis motivates an adaptive sampling strategy that allows on-the-fly identification of and addition of data to regions important for predicting the statistics of interest. Finally, we demonstrate that we can compute accurate statistics for a 75-dimensional model of sudden stratospheric warming. This system provides a stringent test bed for our *** Codes 65C40, 62M20, 62M45 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates

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arXiv 2022年

作者： Bi, Lijuan Cohl, Howard S. Volkmer, Hans Department of Mathematics The Ohio State University at Newark NewarkOH43055 United States Applied and Computational Mathematics Division National Institute of Standards and Technology Mission ViejoCA92694 United States Department of Mathematical Sciences University of Wisconsin-Milwaukee MilwaukeeWI53201-0413 United States

We derive an expansion for the fundamental solution of Laplace's equation in at-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are at-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lamé-Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé-Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates. © 2022, CC0.

关键词： Laplace transforms