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arXiv 2023年

作者： Kapit, Eliot Barton, Brandon A. Feeney, Sean Grattan, George Patnaik, Pratik Sagal, Jacob Carr, Lincoln D. Oganesyan, Vadim Department of Physics Colorado School of Mines 1523 Illinois St GoldenCO80401 United States Department of Applied Mathematics and Statistics Colorado School of Mines 1500 Illinois St GoldenCO80401 United States Quantum Engineering Program Colorado School of Mines 1523 Illinois St GoldenCO80401 United States Department of Physics and Astronomy College of Staten Island CUNY Staten IslandNY10314 United States Physics Program and Initiative for the Theoretical Sciences The Graduate Center CUNY New YorkNY10016 United States Center for Computational Quantum Physics Flatiron Institute 162 5th Avenue New YorkNY10010 United States

A canonical feature of the constraint satisfaction problems in NP is approximation hardness, where in the worst case, finding sufficient-quality approximate solutions is exponentially hard for all known methods. Fundamentally, the lack of any guided local minimum escape method ensures both exact and approximate classical approximation hardness, but the equivalent mechanism(s) for quantum algorithms are poorly understood. For algorithms based on Hamiltonian time evolution, we explore this question through the prototypically hard MAX-3-XORSAT problem class. We conclude that the mechanisms for quantum exact and approximation hardness are fundamentally distinct. We review known results from the literature, and identify mechanisms that make conventional quantum methods (such as Adiabatic Quantum Computing) weak approximation algorithms in the worst case. We construct a family of "spectrally filtered" quantum algorithms that escape these issues, and develop analytical theories for their performance. We show that, for random hypergraphs in the approximation-hard regime, if we define the energy to be E = Nunsat - Nsat, spectrally filtered quantum optimization will return states with E ≤ qmEGS (where EGS is the ground state energy) in sub-quadratic time, where conservatively, qm 0.59. This is in contrast to qm → 0 for the hardest instances with classical searches. We test all of these claims with extensive numerical simulations. We do not claim that this approximation guarantee holds for all possible hypergraphs, though our algorithm’s mechanism can likely generalize widely. These results suggest that quantum computers are more powerful for approximate optimization than had been previously assumed. © 2023, CC BY.

关键词： Constraint satisfaction problems

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A machine learning-based high-precision density functional method for drug-like molecules

Artificial Intelligence Chemistry

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Artificial Intelligence Chemistry 2024年第1期2卷

作者： Jin Xiao YiXiao Chen LinFeng Zhang Han Wang Tong Zhu Shanghai Engineering Research Center of Molecular Therapeutics and New Drug Development Shanghai Frontiers Science Center of Molecule Intelligent Syntheses School of Chemistry and Molecular Engineering East China Normal University Shanghai 200062 P.R. China Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA AI for Science Institute Beijing 100080 P.R. China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 P.R. China NYU-ECNU Center for Computational Chemistry at NYU Shanghai Shanghai 200062 P.R. China Shenzhen Institute of Advanced Technology Chinese Academy of Sciences Shenzhen 518005 P.R. China

In computer-aided drug discovery, accurately determining the structure and properties of drug-like molecules is of utmost importance. This necessitates the use of precise and efficient electronic structure methods. Here, we developed two deep learning-based density functional methods, namely DeePHF and DeePKS, specifically tailored for drug-like molecules. Notably, DeePKS incorporates self-consistency into its framework. With a limited dataset labelled at the CCSD(T)/def2-TZVP level, both models have been able to achieve chemical accuracy in calculating molecular energies and have demonstrated excellent transferability. We anticipate that further advancements in this field will lead to the development of high-quality density functional methods designed specifically for drug discovery purposes. This research showcases the capabilities of deep learning approaches in simplifying the construction complexity associated with traditional DFT methods.

关键词： Machine Learning Density Functional Theory DeePHF DeePKS

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Convergence of deep fictitious play for stochastic differential games

arXiv

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arXiv 2020年

作者： Han, Jiequn Hu, Ruimeng Long, Jihao Department of Mathematics Princeton University PrincetonNJ08544-1000 United States Department of Mathematics Department of Statistics and Applied Probability University of California Santa BarbaraCA93106-3080 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

Stochastic differential games have been used extensively to model agents' competitions in Finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel efficient tool for finding Markovian Nash equilibrium of large N-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into N sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an ϵ-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems. Copyright © 2020, The Authors. All rights reserved.

关键词： Game theory

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DeePN2: A deep learning-based non-Newtonian hydrodynamic model

arXiv

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arXiv 2021年

作者： Fang, Lidong Ge, Pei Zhang, Lei Weinan, E. Lei, Huan Department of Computational Mathematics Science and Engineering Michigan State University MI48824 United States School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing100871 China AI for Science Institute Beijing100080 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Statistics and Probability Michigan State University MI48824 United States

A long standing problem in the modeling of non-Newtonian hydrodynamics of polymeric flows is the availability of reliable and interpretable hydrodynamic models that faithfully encode the underlying micro-scale polymer dynamics. The main complication arises from the long polymer relaxation time, the complex molecular structure and heterogeneous interaction. DeePN2, a deep learning-based non-Newtonian hydrodynamic model, has been proposed and has shown some success in systematically passing the micro-scale structural mechanics information to the macro-scale hydrodynamics for suspensions with simple polymer conformation and bond potential. The model retains a multi-scaled nature by mapping the polymer configurations into a set of symmetry-preserving macro-scale features. The extended constitutive laws for these macro-scale features can be directly learned from the kinetics of their micro-scale counterparts. In this paper, we develop DeePN2 using more complex micro-structural models. We show that DeePN2 can faithfully capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention. © 2021, CC BY.

关键词： Non Newtonian flow

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Flexibility and rigidity in steady fluid motion

arXiv

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arXiv 2020年

作者： Constantin, Peter Drivas, Theodore D. Ginsberg, Daniel Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Stony Brook University Stony BrookNY11794 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Flexibility and rigidity properties of steady (time-independent) solutions of the Euler, Boussinesq and Magnetohydrostatic equations are investigated. Specifically, certain Liouville-type theorems are established which show that suitable steady solutions with no stagnation points occupying a two-dimensional periodic channel, or axisymmetric solutions in (hollowed out) cylinder, must have certain structural symmetries. It is additionally shown that such solutions can be deformed to occupy domains which are themselves small perturbations of the base domain. As application of the general scheme, Arnol'd stable solutions are shown to be structurally stable. Copyright © 2020, The Authors. All rights reserved.

关键词： Rigidity

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Dynamical properties of particulate composites derived from ultradense stealthy hyperuniform sphere packings

arXiv

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arXiv 2025年

作者： Vanoni, Carlo Kim, Jaeuk Steinhardt, Paul J. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Stealthy hyperuniform (SHU) many-particle systems are distinguished by a structure factor that vanishes not only at zero wavenumber (as in "standard" hyperuniform systems) but also across an extended range of wavenumbers near the origin. We generate disordered SHU packings of identical and ‘nonoverlapping’ spheres in d-dimensional Euclidean space using a modified collective-coordinate optimization algorithm that incorporates a soft-core repulsive potential between particles in addition to the standard stealthy pair potential. Compared to SHU packings without soft-core repulsions, these SHU packings are ultradense with packing fractions ranging from 0.67-0.86 for d = 2 and 0.47-0.63 for d = 3, spanning a broad spectrum of structures depending on the stealthiness parameter χ. We consider two-phase media composed of hard particles derived from ultradense SHU packings (phase 2) embedded in a matrix phase (phase 1), with varying stealthiness parameter χ and packing fractions . Our main objective is the estimation of the dynamical physical properties of such two-phase media, namely, the effective dynamic dielectric constant and the time-dependent diffusion spreadability, which is directly related to nuclear magnetic relaxation in fluid-saturated porous media. We show through spreadability that two-phase media derived from ultradense SHU packings exhibit faster interphase diffusion due to the higher packing fractions achievable compared to media obtained without soft-core repulsion. The imaginary part of the effective dynamic dielectric constant of SHU packings vanishes at a small wavenumber, implying perfect transparency for the corresponding wavevectors. While a larger packing fraction yields a smaller transparency interval, we show that it also displays a reduced height of the attenuation peak. We also obtain cross-property relations between transparency characteristics and long-time behavior of the spreadability for such two-phase media, showing that one leads to infor

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Preconditioned BFGS-based uncertainty quantification in elastic full waveform inversion

arXiv

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arXiv 2020年

作者： Liu, Qiancheng Beller, Stephen Lei, Wenjie Peter, Daniel Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ08544 United States Thuwal Saudi Arabia Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Full Waveform Inversion (FWI) has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, uncertainty quantification is related to the inverse Hessian (or the posterior covariance matrix). Even for common geophysical inverse problems its calculation is beyond the computational and storage capacities of the largest high-performance computing systems. In this study, we amend the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to perform uncertainty quantification for large-scale applications. For seismic inverse problems, the limited-memory BFGS (L-BFGS) method prevails as the most efficient quasi-Newton method. We aim to augment it further to obtain an approximate inverse Hessian for uncertainty quantification in FWI. To facilitate retrieval of the inverse Hessian, we combine BFGS (essentially a full-history L-BFGS) with randomized singular value decomposition to determine a low-rank approximation of the inverse Hessian. Setting the rank number equal to the number of iterations makes this solution efficient and memory-affordable even for large-scale problems. Furthermore, based on the Gauss-Newton method, we formulate different initial, diagonal Hessian matrices as preconditioners for the inverse scheme and compare their performances in elastic FWI applications. We highlight our approach with the elastic Marmousi benchmark model, demonstrating the applicability of preconditioned BFGS for large-scale FWI and uncertainty quantification. Copyright © 2020, The Authors. All rights reserved.

关键词： Uncertainty analysis

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Modeling subgrid-scale forces by spatial artificial neural networks in large eddy simulation of turbulence

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Physical Review Fluids 2020年第5期5卷 054606-054606页

作者： Chenyue Xie Jianchun Wang Weinan E Shenzhen Key Laboratory of Complex Aerospace Flows Center for Complex Flows and Soft Matter Research Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China Department of Mathematics Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Spatial artificial neural network (ANN) models are developed for subgrid-scale (SGS) forces in the large eddy simulation (LES) of turbulence. The input features are based on the first-order derivatives of the filtered velocity field at different spatial locations. The correlation coefficients of SGS forces predicted by the spatial artifical neural network (SANN) models with reasonable spatial stencil geometry can be made larger than 0.99 in an a priori analysis, and the relative error of SGS forces can be made smaller than 15%, much smaller than that of the traditional gradient model. In a posteriori analysis, a detailed comparison is made on the results of LES using the SANN model, implicit large eddy simulation (ILES), the dynamic Smagorinsky model (DSM), and the dynamic mixed model (DMM) at grid resolution of 643. It is shown that the SANN model performs better than the ILES, DSM, and DMM models in the prediction of the spectrum and other statistical properties of the velocity field, as well as the instantaneous flow structures. These results suggest that artificial neural network with consideration of spatial characteristics is a very effective tool for developing advanced SGS models in LES of turbulence.

关键词： Compressible flows Turbulence Artificial neural networks

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Deep learning of accurate force field of ferroelectric HfO2

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Physical Review B 2021年第2期103卷 024108-024108页

作者： Jing Wu Yuzhi Zhang Linfeng Zhang Shi Liu Fudan University Shanghai 200433 China School of Science Westlake University Hangzhou Zhejiang 310024 China Institute of Natural Sciences Westlake Institute for Advanced Study Hangzhou Zhejiang 310024 China Yuanpei College Peking University Beijing 100871 China Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Key Laboratory for Quantum Materials of Zhejiang Province Hangzhou Zhejiang 310024 China

The discovery of ferroelectricity in HfO2-based thin films opens up new opportunities for using this silicon-compatible ferroelectric to realize low-power logic circuits and high-density nonvolatile memories. The functional performances of ferroelectrics are intimately related to their dynamic responses to external stimuli such as electric fields at finite temperatures. Molecular dynamics is an ideal technique for investigating dynamical processes on large length and time scales, though its applications to new materials are often hindered by the limited availability and accuracy of classical force fields. Here we present a deep neural network–based interatomic force field of HfO2 learned from ab initio data using a concurrent learning procedure. The model potential is able to predict structural properties such as elastic constants, equation of states, phonon dispersion relationships, and phase transition barriers of various hafnia polymorphs with accuracy comparable with density functional theory calculations. The validity of this model potential is further confirmed by the reproduction of experimental sequences of temperature-driven ferroelectric-paraelectric phase transitions of HfO2 with isobaric-isothermal ensemble molecular dynamics simulations. We suggest a general approach to extend the model potential of HfO2 to related material systems including dopants and defects.

关键词： Interatomic & molecular potentials Structural properties Ferroelectrics Artificial neural networks Molecular dynamics

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Inferring solar differential rotation through normal-mode coupling using bayesian statistics

arXiv

引用

arXiv 2021年

作者： Kashyap, Samarth Ganesh Bharati Das, Srijan Hanasoge, Shravan M. Woodard, Martin F. Tromp, Jeroen Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences Princeton University PrincetonNJ United States NorthWest Research Associates Boulder Office 3380 Mitchell Lane BoulderCO United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global-scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via "a-coefficients". However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analysing eigenfunction corrections, which respect cross coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields a-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer a-coefficients due to differential rotation from cross spectra for both f-modes and p-modes. We demonstrate that this technique works reasonably well for modes with angular degrees ` = 50 − 291. The inferred a3−coefficients are found to be within 1 nHz of the frequency splitting values for ` > 200. We also show that the technique fails at ` © 2021, CC BY.

关键词： Eigenvalues and eigenfunctions

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