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End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems 18

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End-to-end symmetry preserving inter-atomic potential energy...

Proceedings of the 32nd International Conference on Neural Information Processing Systems

作者： Linfeng Zhang Jiequn Han Han Wang Wissam A. Saidi Roberto Car E. Weinan Program in Applied and Computational Mathematics Princeton University Institute of Applied Physics and Computational Mathematics China and CAEP Software Center for High Performance Numerical Simulation China Department of Mechanical Engineering and Materials Science University of Pittsburgh Program in Applied and Computational Mathematics Princeton University and Department of Chemistry and Department of Physics Princeton University and Princeton Institute for the Science and Technology of Materials Princeton University Program in Applied and Computational Mathematics Princeton University and Department of Mathematics Princeton University and Beijing Institute of Big Data Research China

Machine learning models are changing the paradigm of molecular modeling, which is a fundamental tool for material science, chemistry, and computational biology. Of particular interest is the inter-atomic potential energy surface (PES). Here we develop Deep Potential - Smooth Edition (DeepPot-SE), an end-to-end machine learning-based PES model, which is able to efficiently represent the PES of a wide variety of systems with the accuracy of ab initio quantum mechanics models. By construction, DeepPot-SE is extensive and continuously differentiable, scales linearly with system size, and preserves all the natural symmetries of the system. Further, we show that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity.

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Sensitivity kernels for inferring Lorentz stresses from normal-mode frequency splittings in the Sun

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arXiv 2020年

作者： Das, Srijan Bharati Chakraborty, Tuneer Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences and Program in Applied & Computational Mathematics Princeton University Princeton NJ United States

Departures from standard spherically symmetric solar models, in the form of perturbations such as global and local-scale flows and structural asphericities, result in the splitting of eigenfrequencies in the observed spectrum of solar oscillations. Drawing from prevalent ideas in normal-mode coupling theory in geophysical literature, we devise a procedure that enables the computation of sensitivity kernels for general Lorentz stress fields in the Sun. Mode coupling due to any perturbation requires careful consideration of self- and cross-coupling of multiplets. Invoking the isolated-multiplet approximation allows for limiting the treatment to purely self-coupling, requiring significantly less computational resources. We identify the presence of such isolated multiplets under the effect of Lorentz stresses in the Sun. Currently, solar missions allow precise measurements of self-coupling of multiplets via "a-coefficients" and the cross-spectral correlation signal which enables the estimation of the "structure coefficients". We demonstrate the forward problem for both self-coupling (a-coefficients) and cross-coupling (structure coefficients). In doing so, we plot the self-coupling kernels and estimate a-coefficients arising from a combination of deep-toroidal and surface-dipolar axisymmetric fields. We also compute the structure coefficients for an arbitrary general magnetic field (real and solenoidal) and plot the corresponding "splitting function", a convenient way to visualize the splitting of multiplets under 3D internal perturbations. The results discussed in this paper pave the way to formally pose an inverse problem, and infer solar internal magnetic fields. Copyright © 2020, The Authors. All rights reserved.

关键词： Chemical reactions

Community Detection and Stochastic Block Models

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arXiv 2017年

作者： Abbe, Emmanuel Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Princeton United States

The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatorial statistics and more generally data science. This monograph surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational tradeoffs, and for various recovery requirements such as exact, partial and weak recovery. The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal SNR-mutual information tradeoff for partial recovery, and the gap between information-theoretic and computational thresholds. The monograph gives a principled derivation of the main algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, (linearized) belief propagation, classical/nonbacktracking spectral methods and graph powering. Extensions to other block models, such as geometric block models, and a few open problems are also discussed. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

Neural network representation of electronic structure from ab initio molecular dynamics

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Science Bulletin 2022年第1期67卷 29-37,M0003页

作者： Qiangqiang Gu Linfeng Zhang Ji Feng International Center for Quantum Materials School of PhysicsPeking UniversityBeijing 100871China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Collaborative Innovation Center of Quantum Matter Beijing 100871China

Despite their rich information content,electronic structure data amassed at high volumes in ab initio molecular dynamics simulations are generally *** introduce a transferable high-fidelity neural network representation of such data in the form of tight-binding Hamiltonians for crystalline *** predictive representation of ab initio electronic structure,combined with machinelearning boosted molecular dynamics,enables efficient and accurate electronic evolution and *** it is applied to a one-dimension charge-density wave material,carbyne,we are able to compute the spectral function and optical conductivity in the canonical *** spectral functions evaluated during soliton-antisoliton pair annihilation process reveal significant renormalization of low-energy edge modes due to retarded electron-lattice coupling beyond the Born-Oppenheimer *** availability of an efficient and reusable surrogate model for the electronic structure dynamical system will enable calculating many interesting physical properties,paving the way to previously inaccessible or challenging avenues in materials modeling.

关键词： Neural network Tight-binding model Electronic structure ab initio molecular dynamics

Representation formulas and pointwise properties for Barron functions

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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Department of Mathematics Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Princeton University Program in Applied and Computational Mathematics 205 Fine Hall - Washington Road PrincetonNJ08544 United States

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only C1-diffeomorphisms which Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly *** Codes 68T07, 46E15, 26B35, 26B40 Copyright © 2020, The Authors. All rights reserved.

关键词： Network layers

Probabilistic Theory of Mean Field Games with Applications II 1

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丛书名： Probability Theory and Stochastic Modelling

1000年

作者： René Carmona François Delarue

ISBN: (数字)9783319564364

ISBN: (纸本)9783319564357;9783030132590

Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.

关键词： Probability Theory and Stochastic Processes Calculus of Variations and Optimal Control Optimization Partial Differential Equations Economic Theory/Quantitative Economics/Mathematical Methods

The phase diagram of a deep potential water model

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arXiv 2021年

作者： Zhang, Linfeng Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 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 Beijing Institute of Big Data Research Beijing100871 China

Using the Deep Potential methodology, we construct a model that reproduces accurately the potential energy surface of the SCAN approximation of density functional theory for water, from low temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region. The computational efficiency of the model makes it possible to predict its phase diagram using molecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluid phases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predicted correctly, with the exception of ice III and XV that are stable in experiments, but metastable in the model. The evolution of the atomic dynamics upon heating, as ice VII transforms first into ice VII′′ and then into an ionic fluid, reveals that molecular dissociation and breaking of the ice rules coexist with strong covalent fluctuations, explaining why only partial ionization was inferred in experiments. © 2021, CC BY-NC-ND.

关键词： Phase diagrams

AN OPTIMAL SCHEDULED LEARNING RATE FOR A RANDOMIZED KACZMARZ ALGORITHM

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arXiv 2022年

作者： Marshall, Nicholas F. Mickelin, Oscar Department of Mathematics Oregon State University United States Program in Applied and Computational Mathematics Princeton University United States

We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving Ax ≈ b + Ε, where Ax = b is a consistent linear system and Ε has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases;in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-W function of an exponential. Copyright © 2022, The Authors. All rights reserved.

关键词： Linear systems

Nonbacktracking bounds on the influence in independent cascade models

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arXiv 2017年

作者： Abbe, Emmanuel Kulkarni, Sanjeev Lee, Eun Jee Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University Princeton United States Department of Electrical Engineering Princeton University Princeton United States Program in Applied and Computational Mathematics Princeton University Princeton United States

This paper develops upper and lower bounds on the influence measure in a network, more precisely, the expected number of nodes that a seed set can influence in the independent cascade model. In particular, our bounds exploit nonbacktracking walks, Fortuin–Kasteleyn–Ginibre (FKG) type inequalities, and are computed by message passing implementation. Nonbacktracking walks have recently allowed for headways in community detection, and this paper shows that their use can also impact the influence computation. Further, we provide a knob to control the trade-off between the efficiency and the accuracy of the bounds. Finally, the tightness of the bounds is illustrated with simulations on various network models. Copyright © 2017, The Authors. All rights reserved.

关键词： Economic and social effects