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Hyper-molecules: On the representation and recovery of dynamical structures, with application to flexible macro-molecular structures in cryo-EM

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arXiv 2019年

作者： Lederman, Roy R. Andén, Joakim Singer, Amit Department of Statistics and Data Science Yale University New HavenCT United States Center for Computational Mathematics Flatiron Institute New YorkNY United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for determining the 3-D structure of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. The molecular structures are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori;one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in determining rigid structures from homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the "hyper-molecule" framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for recovering such maps of heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a prototype applied to synthetic data. Copyright © 2019, The Authors. All rights reserved.

关键词： Medical imaging

Security Management for Industrial Safety Critical Applications 1

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丛书名： Asset Analytics

1000年

作者： Raj Kamal Kaur Lalit Kumar Singh Pooja Singh Ajit K. Verma

CVIT: CONTINUOUS VISION TRANSFORMER FOR OPERATOR LEARNING

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arXiv 2024年

作者： Wang, Sifan Seidman, Jacob H. Sankaran, Shyam Wang, Hanwen Pappas, George J. Perdikaris, Paris Institution for Foundation of Data Science Yale University United States Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania United States Department of Electrical and Systems Engineering University of Pennsylvania United States

Operator learning, which aims to approximate maps between infinite-dimensional function spaces, is an important area in scientific machine learning with applications across various physical domains. Here we introduce the Continuous Vision Transformer (CViT), a novel neural operator architecture that leverages advances in computer vision to address challenges in learning complex physical systems. CViT combines a vision transformer encoder, a novel grid-based coordinate embedding, and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. This design allows for flexible output representations and consistent evaluation at arbitrary resolutions. We demonstrate CViT’s effectiveness across a diverse range of partial differential equation (PDE) systems, including fluid dynamics, climate modeling, and reaction-diffusion processes. Our comprehensive experiments show that CViT achieves state-of-the-art performance on multiple benchmarks, often surpassing larger foundation models, even without extensive pretraining and roll-out fine-tuning. Taken together, CViT exhibits robust handling of discontinuous solutions, multi-scale features, and intricate spatio-temporal dynamics. Our contributions can be viewed as a significant step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in the physical sciences. All data and code are publicly available at https://***/PredictiveIntelligenceLab/cvit. © 2024, CC BY-NC-SA.

关键词： Computer vision

Tracking origins of below-threshold harmonics with a trajectory-resolved fully quantum approach

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Physical Review A 2016年第6期94卷 063417-063417页

作者： Rui-Hua Xu Yan-Jun Chen Jie Liu Li-Bin Fu National Laboratory of Science and Technology on Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 China Graduate School China Academy of Engineering Physics Beijing 100088 China College of Physics and Information Technology Shaan'xi Normal University Xi'an China HEDPS Center for Applied Physics and Technology Peking University Beijing 100084 China

The emission of below-threshold harmonics (BTHs) is a promising tool for producing vacuum-ultraviolet frequency combs. Nevertheless, the generation mechanism of BTHs is still an open question and a lot of efforts have been devoted to this issue. Here, with a developed trajectory-resolved fully quantum numerical scheme which allows us to “coherently” trace the motion of the rescattering electron, the complex roles of different rescattering trajectories, such as the first return or multiple returns in the origin of BTHs are identified unambiguously. We show that the emission of BTHs is influenced dramatically by laser intensity. At strong laser fields, multiple returns (especially the third return) play an important role in BTHs. At weak fields, effects related to multiphoton processes dominate. The interference of the different contributions to BTHs is also addressed.

关键词： Atomic & molecular processes in external fields High-order harmonic generation

Deep neural network for Wannier function centers

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arXiv 2019年

作者： Zhang, Linfeng Chen, Mohan Wu, Xifan Wang, Han Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States CAPT HEDPS College of Engineering Peking University Beijing100871 China Department of Physics Temple University PhiladelphiaPA19122 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China 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 introduce a deep neural network (DNN) model that assigns the position of the centers of the electronic charge in the snapshots of a molecular dynamics trajectory. The electronic centers are uniquely specified by the unitary transformation that maps the occupied eigenstates onto maximally localized Wannier functions. In combination with deep potential molecular dynamics, a DNN approach to model the atomic potential energy surface at the ab initio density functional level of theory, the scheme makes possible to predict the dielectric properties of insulators for samples and trajectories inaccessible to direct ab initio simulation, without loss of accuracy. As an example, we report calculations of the infrared absorption spectra of light and heavy water at a dispersion inclusive hybrid functional level of theory, finding good agreement with experiment. Extensions to other spectroscopies, like Raman and sum frequency generation, are discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum chemistry

Random-walk shielding-potential viscosity model for warm dense metals

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Physical Review E 2022年第1期106卷 014142-014142页

作者： Yuqing Cheng Haifeng Liu Yong Hou Xujun Meng Qiong Li Yu Liu Xingyu Gao Jianmin Yuan Haifeng Song Jianguo Wang School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100094 China Department of Physics College of Liberal Arts and Sciences National University of Defense Technology Changsha 410073 China Graduate School China Academy of Engineering Physics Beijing 100193 China

We develop a model, called the “random-walk shielding-potential viscosity model” (RWSP-VM) that introduces the statistics of random-walk ions and the Debye shielding effect to describe the viscosities of warm dense metals. The viscosities of several metals with low to high atomic numbers (Be, Al, Fe, and U) are calculated using the analytical expression of RWSP-VM. Additionally, we simulate the viscosities of Fe and Be by employing the Langevin molecular dynamics (MD) and classical MD, while the MD data for Al and U are obtained from a previous work. The results of the RWSP-VM are in good agreement with the MD results, which validates the proposed model. Furthermore, we compare the RWSP-VM with the one-component plasma model and Yukawa viscosity model and show that the three models yield results in excellent agreement with each other in the regime where the RWSP-VM is applicable. These results indicate that the RWSP-VM is a universal, accurate, and highly efficient model for calculating the viscosity of metals in the warm dense state. The code of the proposed RWSP-VM is provided, and it is envisaged that it will have broad application prospects in numerous fields.

关键词： Classical statistical mechanics Classical transport High-energy-density plasmas Random walks Ionic fluids Liquid metals Metals First-principles calculations Molecular dynamics

Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics

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arXiv 2017年

作者： Zhang, Linfeng Han, Jiequn Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China CAEP Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China 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 Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Data Science Beijing International Center for Mathematical Research Peking University Beijing Institute of Big Data Research Beijing100871 China

We introduce a scheme for molecular simulations, the Deep Potential Molecular Dynamics (DeePMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is "first principle-based" in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DeePMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size. Copyright © 2017, The Authors. All rights reserved.

关键词： Molecular dynamics

Structural characterization of many-particle systems on approach to hyperuniform states

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arXiv 2021年

作者： Torquato, Salvatore Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics and biology. The focus of this work is the exploration of quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space Rd approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes as well as the crossover point between them in terms of the "volume" coefficient A and "surface-area" coefficient B associated with the local number variance σ2(R) for a spherical window of radius R. The larger the ratio B/A, the larger the hyperuniform scaling regime, which becomes of infinite extent in the limit B/A → ∞. To complement the known direct-space representation of the coefficient B in terms of the total correlation function h(r), we derive its corresponding Fourier representation in terms of the structure factor S(k), which is especially useful when scattering information is available experimentally or theoretically. We also demonstrate that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio B/A, as well as other diagnostic measures of hyperuniformity, including the hyperuniformity index H and the direct-correlation function length scale ξc, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. Specifically, we analyze equilibrium systems of hard rods and "sticky" hard-sphere systems in arbitrary space dimension d as a function of density. We also examine low-temperature excited states of many-particle systems interacting with "steal

关键词： Temperature

A theory of non-linear feature learning with one gradient step in two-layer neural networks 24

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A theory of non-linear feature learning with one gradient st...

Proceedings of the 41st International Conference on Machine Learning

作者： Behrad Moniri Donghwan Lee Hamed Hassani Edgar Dobriban Department of Electrical and Systems Engineering University of Pennsylvania PA Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PA Department of Statistics and Data Science University of Pennsylvania PA

Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of gradient descent on the first layer can lead to feature learning; characterized by the appearance of a separated rank-one component--spike--in the spectrum of the feature matrix. However, with a constant gradient descent step size, this spike only carries information from the linear component of the target function and therefore learning non-linear components is impossible. We show that with a learning rate that grows with the sample size, such training in fact introduces multiple rank-one components, each corresponding to a specific polynomial feature. We further prove that the limiting largedimensional and large sample training and test errors of the updated neural networks are fully characterized by these spikes. By precisely analyzing the improvement in the training and test errors, we demonstrate that these non-linear features can enhance learning.

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