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Deep density: Circumventing the kohn-sham equations via symmetry preserving neural networks

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arXiv 2019年

作者： Zepeda-Núñez, Leonardo Chen, Yixiao Zhang, Jiefu Zhang, Linfeng Jia, Weile Lin, Lin Department of Mathematics University of Wisconsin-Madison MadisonWI53706 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics University of California Berkeley BerkeleyCA94720 United States Department of Mathematics University of California Berkeley Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

The recently developed Deep Potential [Phys. Rev. Lett. 120, 143001, 2018] is a powerful method to represent general inter-atomic potentials using deep neural networks. The success of Deep Potential rests on the proper treatment of locality and symmetry properties of each component of the network. In this paper, we leverage its network structure to effectively represent the mapping from the atomic configuration to the electron density in Kohn-Sham density function theory (KS-DFT). By directly targeting at the self-consistent electron density, we demonstrate that the adapted network architecture, called the Deep Density, can effectively represent the electron density as the linear combination of contributions from many local clusters. The network is constructed to satisfy the translation, rotation, and permutation symmetries, and is designed to be transferable to different system sizes. We demonstrate that using a relatively small number of training snapshots, Deep Density achieves excellent performance for one-dimensional insulating and metallic systems, as well as systems with mixed insulating and metallic characters. We also demonstrate its performance for real three-dimensional systems, including small organic molecules, as well as extended systems such as water (up to 512 molecules) and aluminum (up to 256 atoms). Copyright © 2019, The Authors. All rights reserved.

关键词： Network architecture

On the cross-validation bias due to unsupervised preprocessing

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arXiv 2019年

作者： Moscovich, Amit Rosset, Saharon Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Statistics and Operations Research Tel-Aviv University Tel-Aviv Israel

Cross-validation is the de facto standard for predictive model evaluation and selection. In proper use, it provides an unbiased estimate of a model's predictive performance. However, data sets often undergo various forms of data-dependent preprocessing, such as mean-centering, rescaling, dimensionality reduction, and outlier removal. It is often believed that such preprocessing stages, if done in an unsupervised manner (that does not incorporate the class labels or response values) are generally safe to do prior to cross-validation. In this paper, we study three commonly-practiced preprocessing procedures prior to a regression analysis: (i) variance-based feature selection;(ii) grouping of rare categorical features;and (iii) feature rescaling. We demonstrate that unsupervised preprocessing can, in fact, introduce a substantial bias into cross-validation estimates and potentially hurt model selection. This bias may be either positive or negative and its exact magnitude depends on all the parameters of the problem in an intricate manner. Further research is needed to understand the real-world impact of this bias across different application domains, particularly when dealing with small sample sizes and high-dimensional *** Codes 62-07 Copyright © 2019, The Authors. All rights reserved.

关键词： Regression analysis

Dynamic Mode Decomposition and Sparse Measurements for Characterization and Monitoring of Power System Disturbances

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arXiv 2019年

作者： Ramos, J. Jorge Kutz, J. Nathan Graduate Program in Electrical Engineering Cinvestav Unidad Guadalajara Guadalajaram Jal Mexico Department of Applied Mathematics University of Washington SeattleWA98195 United States

We introduce the dynamics mode decomposition for monitoring wide-area power grid networks from sparse measurement data. The mathematical framework fuses data from multiple sensors based on multivariate statistics, providing accurate full state estimation from limited measurements and generating data-driven forecasts for the state of the system. Our proposed data-driven strategy, which is based on energy metrics, can be used for the analysis of major disturbances in the network. The approach is tested and validated using time domain simulations in the IEEE 118 bus system under various disturbance scenarios and under different sparse observations of the system. In addition to state reconstruction, the minimal number of sensors required for monitoring disturbances can be evaluated. Visualization techniques are developed in order to aid in the analysis and characterization of the system after disturbance. Copyright © 2019, The Authors. All rights reserved.

关键词： Time domain analysis

Isotope effects on molecular structures and electronic properties of liquid water via deep potential molecular dynamics based on SCAN functional

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arXiv 2020年

作者： Xu, Jianhang Zhang, Chunyi Zhang, Linfeng Chen, Mohan Santra, Biswajit Wu, Xifan Department of Physics Temple University PhiladelphiaPA19122 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States CAPT HEDPS College of Engineering Peking University Beijing100871 China Institute for Computational Molecular Science Temple University PhiladelphiaPA19122 United States

Feynman path-integral deep potential molecular dynamics (PI-DPMD) calculations have been employed to study both light (H2O) and heavy water (D2O) within the isothermal–isobaric ensemble. In particular, the deep neural network is trained based on ab initio data obtained from the strongly constrained and appropriately normed (SCAN) exchange-correlation functional. Because of the lighter mass of hydrogen than deuteron, the properties of light water is more influenced by nuclear quantum effect than those of heavy water. Clear isotope effects are observed and analyzed in terms of hydrogen-bond structure and electronic properties of water that are closely associated with experimental observables. The molecular structures of both liquid H2O and D2O agree well with the data extracted from scattering experiments. The delicate isotope effects on radial distribution functions and angular distribution functions are well reproduced as well. Our approach demonstrates that deep neural network combined with SCAN functional based ab initio molecular dynamics provides an accurate theoretical tool for modeling water and its isotope effects. Copyright © 2020, The Authors. All rights reserved.

关键词： Electronic properties

Effective Elastic Wave Characteristics of Composite Media

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arXiv 2020年

作者： Kim, Jaeuk Torquato, S. 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

We derive exact expressions for effective elastodynamic properties of two-phase composites in the long-wavelength (quasistatic) regime via homogenized constitutive relations that are local in space. This is accomplished by extending the "strong-contrast" expansion formalism that was previously applied to the static problem. These strong-contrast expansions explicitly incorporate complete microstructural information of the composite via an infinite set of n-point correlation functions. Utilizing the rapid-convergence properties of these series expansions (even for extreme contrast ratios), we extract accurate approximations that depend on the microstructure via the spectral density, which is easy to compute or measure for any composite. We also investigate the predictive power of modifications of such approximation formulas postulated elsewhere [J. Kim and S. Torquato, Proc. Nat. Acad. Sci. 117, 8764 (2020)] to extend their applicability beyond the quasistatic regime. The accuracy of these nonlocal microstructure-dependent approximations is validated by comparison to full-waveform simulation results for certain models of dispersions. We apply our formulas to a variety of models of nonhyperuniform and hyperuniform disordered composites. We demonstrate that hyperuniform systems are less lossy than their nonhyperuniform counterparts in the quasistatic regime, and stealthy hyperuniform media can be perfectly transparent for a wide range of wavenumbers. Finally, we discuss how to utilize our approximations for engineering composites with prescribed elastic wave characteristics. Copyright © 2020, The Authors. All rights reserved.

关键词： Microstructure

Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field

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arXiv 2020年

作者： Bose, Amartya Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics 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

Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems like liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such systems involving spin chains are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform extensive analysis of these systems using density matrix renormalization group (DMRG) to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the "τ order metric" of the ground state, the measure of hyperuniformity and the second cumulant of the total magnetization along the x-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of transverse magnetic field may remarkably lead to increase in the order of the ground state as measured by the "τ order metric," even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping on to a system of spinless ferm

关键词： Phase transitions

Nonlocal effective electromagnetic wave characteristics of composite media: Beyond the quasistatic regime

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arXiv 2020年

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

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor Εe(kI, ω) for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and kI are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0 ≤ |kI|. 1 (where is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for Εe(kI, ω) that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperunif

关键词： Microstructure

Coarse-scale PDEs from fine-scale observations via machine learning

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arXiv 2019年

作者： Lee, Seungjoon Kooshkbaghi, Mahdi Spiliotis, Konstantinos Siettos, Constantinos I. Kevrekidis, Ioannis G. Department of Chemical and Biomolecular Engineering Johns Hopkins University Program in Applied and Computational Mathematics Princeton University Institute of Mathematics University of Rostock Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Universitá degli Studi di Napoli Federico II

Complex spatiotemporal dynamics of physicochemical processes are often modeled at a microscopic level (through e.g. atomistic, agent-based or lattice models) based on first principles. Some of these processes can also be successfully modeled at the macroscopic level using e.g. partial differential equations (PDEs) describing the evolution of the right few macroscopic observables (e.g. concentration and momentum fields). Deriving good macroscopic descriptions (the so-called "closure problem") is often a time-consuming process requiring deep understanding/intuition about the system of interest. Recent developments in data science provide alternative ways to effectively extract/learn accurate macroscopic descriptions approximating the underlying microscopic observations. In this paper, we introduce a data-driven framework for the identification of unavailable coarse-scale PDEs from microscopic observations via machine learning algorithms. Specifically, using Gaussian Processes, Artificial Neural Networks, and/or Diffusion Maps, the proposed framework uncovers the relation between the relevant macroscopic space fields and their time evolution (the right-hand-side of the explicitly unavailable macroscopic PDE). Interestingly, several choices equally representative of the data can be discovered. The framework will be illustrated through the data-driven discovery of macroscopic, concentration-level PDEs resulting from a fine-scale, Lattice Boltzmann level model of a reaction/transport process. Once the coarse evolution law is identified, it can be simulated to produce long-term macroscopic predictions. Different features (pros as well as cons) of alternative machine learning algorithms for performing this task (Gaussian Processes and Artificial Neural Networks), are presented and discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Neural networks

Nearest-neighbor functions for disordered stealthy hyperuniform many-particle systems

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arXiv 2020年

作者： Middlemas, Timothy M. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics 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

Disordered stealthy many-particle systems in d-dimensional Euclidean space Rd are exotic amorphous states of matter that suppress any single scattering events for a finite range of wavenumbers around the origin in reciprocal space. They are currently the subject of intense fundamental and practical interest. We derive analytical formulas for the nearest-neighbor functions of disordered stealthy many-particle systems. First, we analyze asymptotic small-r approximations and expansions of the nearest-neighbor functions based on the pseudo-hard-sphere ansatz. We then consider the problem of determining how many of the standard n-point correlation functions are needed to determine the nearest neighbor functions, and find that a finite number suffice. Via theoretical and computational methods, we are able to compare the large-r behavior of these functions for disordered stealthy systems to those belonging to crystalline lattices. Such ordered and disordered stealthy systems have bounded hole sizes, and thus compact support for their nearest-neighbor functions. However, we find that the approach to the critical-hole size can be quantitatively different, emphasizing the importance of hole statistics in distinguishing ordered and disordered stealthy configurations. We argue that the probability of finding a hole close to the critical-hole size should decrease as a power law with an exponent only dependent on the space dimension d for ordered systems, but that this probability decays asymptotically faster for disordered systems, with either an increase in the exponent of the power law or a crossover into a decay faster than any power law. This implies that holes close to the critical-hole size are rarer in disordered systems. The rarity of observing large holes in disordered systems creates substantial numerical difficulties in sampling the nearest neighbor distributions near the critical-hole size. This motivates both the need for new computational methods for efficient samp

关键词： Computational methods