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Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions

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arXiv 2017年

作者： Lederman, Roy R. Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely;the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems. Copyright © 2017, The Authors. All rights reserved.

关键词： Inverse problems

Coarse-grained spectral projection (CGSP): A deep learning-assisted approach to quantum unitary dynamics

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arXiv 2020年

作者： Xie, Pinchen Weinan, E. Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We propose the coarse-grained spectral projection method (CGSP), a deep learning-assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show CGSP can extract spectral components of many-body quantum states systematically with sophisticated neural network quantum ansatz. CGSP exploits fully the linear unitary nature of the quantum dynamics, and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on 1D XXZ models with periodic boundary condition are carried out to demonstrate the practicality of CGSP. Copyright © 2020, The Authors. All rights reserved.

关键词： Dynamics

THREE-DIMENSIONAL QUANTUM GRAVITY AS DYNAMICAL TRIANGULATION

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Modern Physics Letters A 1991年第20期6卷 1863-1884页

作者： M. E. AGISHTEIN A. A. MIGDAL Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544–1000 USA Physics Department and Program in Applied and Computational Mathematics Fine Hall Princeton University Princeton NJ 08544-1000 USA

The dynamical triangulation model of 3-dimensional Quantum Gravity is defined and studied. We propose two different algorithms for numerical simulations, leading to consistent results. One is the 3-dimensional generalization of the bonds flip, another is more sophisticated algorithm, based on Schwinger–Dyson equations. We found such care necessary, because our results appear to be quite unexpected. We simulated up to 60000 tetrahedra and observed none of the feared pathologies like factorial growth of the partition function with volume, or collapse to the branched polymer phase. The volume of the Universe grows exponentially when the bare cosmological constant λ approaches the critical value λ c from above, but the closed Universe exists and has peculiar continuum limit. The Universe compressibility diverges as (λ − λ c ) −2 and the bare Newton constant linearly approaches negative critical value as λ goes to λ c , provided the average curvature is kept at zero. The fractal properties turned out to be the same, as in two dimensions, namely the effective Hausdorff dimension grows logarithmically with the size of the test geodesic sphere.

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KOLMOGOROV WIDTH DECAY AND POOR APPROXIMATORS IN MACHINE LEARNING: SHALLOW NEURAL NETWORKS, RANDOM FEATURE MODELS AND NEURAL TANGENT KERNELS

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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544

We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor L2-approximators for the class of two-layer neural networks in high dimension, and that two-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the *** Codes 68T07, 41A30, 41A65, 46E15, 46E22 Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath

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Physical Review B 2007年第10期76卷 104107-104107页

作者： Xiantao Li Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

This paper presents a systematic approach for finding efficient boundary conditions for molecular dynamics simulations of crystalline solids. These boundary conditions effectively eliminate phonon reflection at the boundary and at the same time allow the thermal energy from the bath to be introduced to the system. Our starting point is the Mori-Zwanzig formalism [R. Zwanzig, J. Chem. Phys. 32, 1173 (1960); in Systems Far from Equilibrium, edited by L. Garrido (Interscience, New York, 1980); H. Mori, Prog. Theor. Phys. 33, 423 (1965)] for eliminating the thermal bath, but we take the crucial next step that goes beyond this formalism in order to obtain memory kernels that decay faster. An equivalent variational formulation allows us to find the optimal approximate boundary conditions, after specifying the spatial-temporal domain of dependence for the positions of the boundary atoms. Application to a one-dimensional chain, a two-dimensional Lennard-Jones system, and a three-dimensional model of α-iron with embedded atom potential is presented to demonstrate the effectiveness of this approach.

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Generalization and memorization: The Bias Potential Model

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arXiv 2020年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University Department of Mathematics Princeton University

Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or *** Codes 68T07, 60-08. Copyright © 2020, The Authors. All rights reserved.

关键词： Probability distributions

Integrating machine learning with physics-based modeling

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arXiv 2020年

作者： Weinan, E. Han, Jiequn Zhang, Linfeng Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University

Machine learning is poised as a very powerful tool that can drastically improve our ability to carry out scientific research. However, many issues need to be addressed before this becomes a reality. This article focuses on one particular issue of broad interest: How can we integrate machine learning with physics-based modeling to develop new interpretable and truly reliable physical models? After introducing the general guidelines, we discuss the two most important issues for developing machine learning-based physical models: Imposing physical constraints and obtaining optimal datasets. We also provide a simple and intuitive explanation for the fundamental reasons behind the success of modern machine learning, as well as an introduction to the concurrent machine learning framework needed for integrating machine learning with physics-based modeling. Molecular dynamics and moment closure of kinetic equations are used as examples to illustrate the main issues discussed. We end with a general discussion on where this integration will lead us to, and where the new frontier will be after machine learning is successfully integrated into scientific modeling. Copyright © 2020, The Authors. All rights reserved.

关键词： Molecular dynamics

Theory of differential approximations of radiative transfer equation 1

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作者： Han, Weimin Eichholz, Joseph A. Sheng, Qiwei Department of Mathematics and Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242 United States Department of Mathematics Rose-Hulman Institute of Technology Terre Haute IN 47803 United States

ISBN: (数字)9781461463931

ISBN: (纸本)9781461463924

The radiative transfer equation (RTE) arises in a variety of applications. The equation is challenging to solve numerically for a couple of reasons: high dimensionality, integro-differential form, highly forward-peaked scattering in application. In the literature, various approximations of RTE have been proposed in the literature. In an earlier publication, we explored a family of differential approximations to RTE, to be called RT/DA equations. In this paper, we study the RT/DA equations and investigate numerically the closeness of solutions of the RT/DA equations to that of the RTE. © Springer Science+Business Media New York 2013.

关键词： Radiative transfer

Robust multi-object matching via iterative reweighting of the graph connection Laplacian

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arXiv 2020年

作者： Shi, Yunpeng Li, Shaohan Lerman, Gilad Program in Applied and Computational Mathematics Princeton University School of Mathematics University of Minnesota

We propose an efficient and robust iterative solution to the multi-object matching problem. We first clarify serious limitations of current methods as well as the inappropriateness of the standard iteratively reweighted least squares procedure. In view of these limitations, we suggest a novel and more reliable iterative reweighting strategy that incorporates information from higher-order neighborhoods by exploiting the graph connection Laplacian. We demonstrate the superior performance of our procedure over state-of-the-art methods using both synthetic and real *** Codes 90C26, 90C10, 90C17, 68Q87, 65C20 Copyright © 2020, The Authors. All rights reserved.

关键词： Laplace transforms