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arXiv 2021年

作者： Wang, Qingcan Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States

We propose a unified framework that extends the inference methods for classical hidden Markov models to continuous settings, where both the hidden states and observations occur in continuous time. Two different settings are analyzed: hidden jump process with a finite state space, and hidden diffusion process with a continuous state space. For each setting, we first estimate the hidden states given the observations and model parameters, showing that the posterior distribution of the hidden states can be described by differential equations in continuous time. We then consider the estimation of unknown model parameters, deriving the continuous-time formulas for the expectation-maximization algorithm. We also propose a Monte Carlo method based on the continuous formulation, sampling the posterior distribution of the hidden states and updating the parameter *** Codes 60J25, 65C40 Copyright © 2021, The Authors. All rights reserved.

关键词： Continuous time systems

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Coarse-graining the dynamics of a driven interface in the presence of mobile impurities: Effective description via diffusion maps

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Physical Review E 2009年第3期80卷 031102-031102页

作者： Benjamin E. Sonday Mikko Haataja Ioannis G. Kevrekidis []Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Developing effective descriptions of the microscopic dynamics of many physical phenomena can both dramatically enhance their computational exploration and lead to a more fundamental understanding of the underlying physics. Previously, an effective description of a driven interface in the presence of mobile impurities, based on an Ising variant model and a single empirical coarse variable, was partially successful [M. Haataja et al., Phys. Rev. Lett. 92, 160603 (2004)]; yet it underlined the necessity of selecting additional coarse variables in certain parameter regimes. In this paper we use a data mining approach to help identify the coarse variables required. We discuss the implementation of this diffusion map approach, the selection of a similarity measure between system snapshots required in the approach, and the correspondence between empirically selected and automatically detected coarse variables. We conclude by illustrating the use of the diffusion map variables in assisting the atomistic simulations and we discuss the translation of information between fine and coarse descriptions using lifting and restriction operators.

关键词： Data mining

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Anomalous suppression of large-scale density fluctuations in classical and quantum spin liquids

arXiv

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arXiv 2025年

作者： Chen, Duyu Samajdar, Rhine Jiao, Yang Torquato, Salvatore Materials Research Laboratory University of California Santa BarbaraCA93106 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Center for Theoretical Science 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 Chemistry Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics PrincetonNJ08544 United States

Classical spin liquids (CSLs) are intriguing states of matter that do not exhibit long-range magnetic order and are characterized by an extensive ground-state degeneracy. Adding quantum fluctuations, which induce dynamics between these different classical ground states, can give rise to quantum spin liquids (QSLs). QSLs are highly entangled quantum phases of matter characterized by fascinating emergent properties, such as fractionalized excitations and topological order. One such exotic quantum liquid is the Z2 QSL, which can be regarded as a resonating valence bond (RVB) state formed from superpositions of dimer coverings of an underlying lattice. In this work, we unveil a hidden large-scale structural property of archetypal CSLs and QSLs known as hyperuniformity, i.e., normalized infinite-wavelength density fluctuations are completely suppressed in these systems. In particular, we first demonstrate that classical ensembles of close-packed dimers and their corresponding quantum RVB states are perfectly hyperuniform in general. Subsequently, we focus on a ruby-lattice spin liquid that was recently realized in a Rydberg-atom quantum simulator, and show that the QSL remains effectively hyperuniform even in the presence of a finite density of spinon and vison excitations, as long as the dimer constraint is still largely preserved. Moreover, we demonstrate that metrics based on the framework of hyperuniformity can be used to distinguish the QSL from other proximate quantum phases. These metrics can help identify potential QSL candidates, which can then be further analyzed using more advanced, computationally-intensive quantum numerics to confirm their status as true QSLs. © 2025, CC BY-NC-ND.

关键词： Quantum entanglement

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Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence

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Physical Review E 2019年第5期99卷 053113-053113页

作者： Chenyue Xie Jianchun Wang Ke Li Chao Ma The Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

A subgrid-scale (SGS) model for large-eddy simulation (LES) of compressible isotropic turbulence is constructed by using a data-driven framework. An artificial neural network (ANN) based on local stencil geometry is employed to predict the unclosed SGS terms. The input features are based on the first-order and second-order derivatives of filtered velocity and temperature which appear in the second-order Taylor approximation of the SGS stress and heat flux. It is shown that the proposed ANN-7 model performs better than the gradient model in the a priori test. The correlation coefficient is larger and the relative error is smaller for ANN-7 model as compared to those of the gradient model in the a priori test. In an a posteriori analysis, the performance of ANN-7 model shows advantage over the dynamic Smagorinsky model and dynamic mixed model in the prediction of spectra and structure functions of velocity and temperature, and instantaneous flow structures. Artificial neural network is a promising tool for understanding the physical fundamentals of SGS unclosed terms with further improvement.

关键词： Compressible flows Turbulence

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Traveling-wave solutions to thin-film equations

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Physical Review E 1993年第6期48卷 4423-4423页

作者： Stefanella Boatto Leo P. Kadanoff Piero Olla Computational and Applied Mathematics Program Ryerson Laboratory The University of Chicago Chicago Illinois 60637

Thin films can be effectively described by the lubrication approximation, in which the equation of motion is ht+(hnhxxx)x=0. Here h is a necessarily positive quantity which represents the height or thickness of the film. Different values of n, especially 1, 2, and 3 correspond to different physical situations. This equation permits solutions in the form of traveling disturbances with a fixed form. If u is the propagation velocity, the resulting equation for the disturbance is uhx=(hnhxxx)x. Here, quantitative and qualitative solutions to the equation are presented. The study has been limited to the intervals in x where the solutions are positive. It is found that transitions between different qualitative behaviors occur at n=3, 2, 3/2, and 1/2. For example, if u is not zero, solitonlike solutions defined on a finite interval are only possible for n<3. More specific results can be obtained. In the case in which the velocity is zero, solitons occur for n<2. For n=1, the region 3/21/2, single-minimum solutions diverging at ±∞ are possible. The generic solution, present for all positive values of n, is a receding front, which diverges at finite x for n<0.

关键词：

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Heterogeneous Multiscale Methods: A Review

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Communications in computational Physics 2007年第3期2卷 367-450页

作者： Weinan E Bjorn Engquist Xiantao Li Weiqing Ren Eric Vanden-Eijnden Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Department of Mathematics The University of Texas at AustinAustinTX 78712USA Department of Mathematics Pennsylvania State UniversityUniversity ParkPA 16802USA Department of Mathematics Courant Institute of Mathematical SciencesNew York UniversityNew YorkNY 10012USA.

This paper gives a systematic introduction to HMM,the heterogeneous multiscale methods,including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular *** is illustrated by examples from several application areas,including complex fluids,micro-fluidics,solids,interface problems,stochastic problems,and statistically self-similar *** is given to the technical tools,such as the various constrained molecular dynamics,that have been developed,in order to apply HMM to these *** of mathematical results on the error analysis of HMM are *** review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

关键词： Multi-scale modeling heterogeneous multi-scale method multi-physics models constrained micro-scale solver data estimation.

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Translation of a sphere towards a rigid plane in an Oldroyd-B fluid

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Physical Review Fluids 2024年第8期9卷 083303页

作者： Tachin Ruangkriengsin Rodolfo Brandão Bimalendu Mahapatra Evgeniy Boyko Howard A. Stone Program in Applied and Computational Mathematics Department of Mechanical and Aerospace Engineering Faculty of Mechanical Engineering

We analyze the low-Reynolds-number translation of a sphere towards or away from a rigid plane in an Oldroyd-B fluid under two scenarios: prescribing the sphere's translational velocity, and prescribing the force on the sphere. Leveraging the lubrication approximation and a perturbation expansion in powers of the Deborah number, we develop a comprehensive theoretical analysis that yields analytical approximations for velocity fields, pressures, and forces acting on the sphere. Our framework aids in understanding temporal microstructural changes as the particle-wall gap evolves over time. In particular, we show that alterations in the polymer conformation tensor in response to geometric changes induce additional forces on the sphere. For cases with prescribed velocity, we present a theoretical approach for calculating resistive forces at any order in the Deborah number and utilize a reciprocal theorem to obtain higher-order corrections based on velocity fields in the previous orders. When the sphere translates with a constant velocity, the fluid viscoelasticity decreases the resistive force at the first order. However, at the second-order correction, the direction of the sphere's movement determines whether viscoelasticity increases or decreases the resistive force. For cases with prescribed force, we show that understanding the influence of viscoelasticity on the sphere's translational velocity necessitates a more intricate analysis even at low Deborah numbers. Specifically, we introduce an ansatz for constant force scenarios, and we derive solution forms for general prescribed forces using the method of multiple scales. We find that when a sphere undergoes sedimentation due to its own weight, the fluid viscoelasticity results in a slower settling process, reducing the leading-order sedimentation rate.

关键词： Complex fluids Low Reynolds number flows Lubrication theory Polymer conformation changes Rheology

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The replicator dynamics for multilevel selection in evolutionary games

arXiv

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arXiv 2018年

作者： Cooney, Daniel B. Program in Applied and Computational Mathematics Princeton University

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner’s Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner’s Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady-states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady-state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and *** Codes 92D25, 91A22, 35F20 Copyright © 2018, The Authors. All rights reserved.

关键词： Probability distributions

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Fire sale in financial networks

Fire sale in financial networks

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Annual Conference on Information Sciences and Systems (CISS)

作者： Haoshu Tian E Weinan Program in Applied and Computational Mathematics Princeton University

The default of one bank can cause other banks to default through two channels: financial contagion in the inter-bank liability network and fire sale in the asset selling market. When the defaulted bank cannot fully pay its debt, the loss is transmitted to other banks. When banks rush to sell the same asset simultaneously, they may fall into a Nash equilibrium in which banks compete for liquidity and sell their assets at an artificially low price. In this paper, a model that incorporates these two channels is developed and analyzed theoretically. An algorithm for finding the state in which both the inter-bank liability network and the market are in equilibrium is proposed and tested.

关键词： Vectors Nash equilibrium Fires Educational institutions Propagation losses Macroeconomics

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Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemmas

arXiv

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arXiv 2019年

作者： Cooney, Daniel B. Program in Applied and Computational Mathematics Princeton University

Here we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag-Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the Prisoners’ Dilemma and Hawk-Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: if groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamics

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