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Sharp rates of convergence for accumulated spectrograms

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arXiv 2017年

作者： Abreu, Luis Daniel Pereira, Joao Romero, Jose Luis Acoustics Research Institute Austrian Academy of Science Wohllebengasse 12-14 ViennaA-1040 Austria Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent *** Codes 65F18, 94A12, 65N21, 42C25 Copyright © 2017, The Authors. All rights reserved.

关键词： Spectrographs

Artificial neural network approach for turbulence models: A local framework

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arXiv 2021年

作者： Xie, Chenyue Xiong, Xiangming Wang, Jianchun Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen518055 China

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by artificial neural network (ANN) based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model for the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN and Spalart-Allmaras (SA) models. It is shown that the LANN model performs better than the SA model in the prediction of the average velocity, wall-shear stress and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation (DNS) data. The LANN model trained in low Reynolds number Re = 2800 can be directly applied in the cases of high Reynolds numbers Re = 5600, 10595, 19000, 37000 with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved *** Codes 76F40, 76F55, 76N99 Copyright © 2021, The Authors. All rights reserved.

关键词： Neural networks

NOMAD: Nonlinear Manifold Decoders for Operator Learning

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arXiv 2022年

作者： Seidman, Jacob H. Kissas, Georgios Perdikaris, Paris Pappas, George J. 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

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost. © 2022, CC BY-NC-SA.

关键词： Benchmarking

Estimation in the group action channel

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arXiv 2018年

作者： Abbe, Emmanuel Pereira, João M. Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Electrical Engineering Department Princeton University PrincetonNJ United States Department of Mathematics Princeton University PrincetonNJ United States

MSC Codes 94A15, 62B10We analyze the problem of estimating a signal from multiple measurements on a group action channel that linearly transforms a signal by a random group action followed by a fixed projection and additive Gaussian noise. This channel is motivated by applications such as multi-reference alignment and cryoelectron microscopy. We focus on the large noise regime prevalent in these applications. We give a lower bound on the mean square error (MSE) of any asymptotically unbiased estimator of the signal's orbit in terms of the signal's moment tensors, which implies that the MSE is bounded away from 0 when N=σ2dis bounded from above, where N is the number of observations, σ is the noise standard deviation, and d is the so-called moment order cutoff. In contrast, the maximum likelihood estimator is shown to be consistent if N=σ2ddiverges. Copyright © 2018, The Authors. All rights reserved.

关键词： Mean square error

Multireference alignment is easier with an aperiodic translation distribution

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arXiv 2017年

作者： Abbe, Emmanuel Bendory, Tamir Leeb, William Pereira, João M. Sharon, Nir Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Electrical Engineering Department Princeton University PrincetonNJ United States Department of Mathematics Princeton University PrincetonNJ United States

In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (the signal energy divided by the noise variance) in order to drive the mean-square error (MSE) to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as ω(1/SNR3). In this work, using a generalization of the Chapman-Robbins bound for orbits and expansions of the X2divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as ω(1/SNR2). This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on non-convex optimization and expectation-maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model. Copyright © 2017, The Authors. All rights reserved.

关键词： Maximum principle

Deep Potentials for Materials Science

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arXiv 2022年

作者： Wen, Tongqi Zhang, Linfeng Wang, Han Weinan, E. Srolovitz, David J. Department of Mechanical Engineering The University of Hong Kong Hong Kong DP Technology Beijing China AI for Science Institute Beijing China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing China HEDPS CAPT College of Engineering Peking University Beijing China School of Mathematical Sciences Peking University Beijing China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton United States Shenzhen China

To fill the gap between accurate (and expensive) ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials, a new class of descriptions of atomic interactions has emerged and been widely applied;i.e., machine learning potentials (MLPs). One recently developed type of MLP is the Deep Potential (DP) method. In this review, we provide an introduction to DP methods in computational materials science. The theory underlying the DP method is presented along with a step-by-step introduction to their development and use. We also review materials applications of DPs in a wide range of materials systems. The DP Library provides a platform for the development of DPs and a database of extant DPs. We discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials. Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

DeePKS Model for Halide Perovskites with the Accuracy of Hybrid Functional

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arXiv 2023年

作者： Ou, Qi Tuo, Ping Li, Wenfei Wang, Xiaoxu Chen, Yixiao Zhang, Linfeng AI for Science Institute Beijing100080 China DP Technology Beijing100080 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Accurate prediction for the electronic structure properties of halide perovskites plays a significant role in the design of highly efficient and stable solar cells. While density functional theory (DFT) within the generalized gradient approximation (GGA) offers reliable prediction in terms of lattice constants and potential energy surface for halide perovskites, it severely underestimates the band gap due to the lack of non-local exact exchange term, which exists in computationally expensive hybrid functionals. In this work, a universal Deep Kohn-Sham (DeePKS) model based on neural network is trained so as to enable electronic structure calculations with the accuracy of hybrid functional HSE06 and the efficiency comparable to GGA functional, for a plethora of halide perovskites, i.e., ABX3 (A=FA, MA, Cs;B=Sn, Pb;X=Cl, Br, I). Forces, band gaps, and density of states (DOS) predicted by our DeePKS model for all aforementioned perovskites are in good agreement with the HSE06 results, with significantly improved efficiency. In addition, even though the spin-orbit coupling (SOC) effect has not been taken into consideration during the training process, DeePKS+SOC offers highly consistent band gap and DOS as compared to HSE06+SOC for Pb-containing systems. We believe such DeePKS model can be readily applied for an accurate yet efficient prediction of various properties for the family of halide perovskites. © 2023, CC BY.

关键词： Energy gap

A class of dimension-free metrics for the convergence of empirical measures

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arXiv 2021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 The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics (e.g., the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables;2. The convergence of n-particle system to the solution to McKean-Vlasov stochastic differential equation;3. The construction of an Ε-Nash equilibrium for a homogeneous n-player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without *** Codes 60B10, 60E15, 60K35, 91A16, 60H10 © 2021, CC BY.

关键词： Stochastic systems

Spectral Dynamics of Spatially Incoherent Modulation Instability

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Physical Review Letters 2012年第26期108卷 263902-263902页

作者： Can Sun Laura Waller Dmitry V. Dylov Jason W. Fleischer Department of Electrical Engineering Princeton University Princeton New Jersey 08544 USA General Electric Global Research Center Niskayuna New York 12309 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

To date, all experiments in nonlinear statistical optics have relied on beams whose transverse spatial statistics were Gaussian. Here, we present a new technique to generalize these studies by using a spatial light modulator to create spatially incoherent beams with arbitrary spectral distributions. As a specific example of the new dynamics possible, we consider the spatial modulation instability of a partially coherent beam. We show that, for statistical beams of uniform intensity and equal correlation length, the underlying spectral shape determines the threshold and visibility of intensity modulations as well as the spectral profile of the growing sidebands. We demonstrate the behavior using statistical light, but the results will hold for any wave-kinetic system, such as plasma, ultracold gases, and turbulent acoustic waves.

关键词： NONLINEAR optics MODULATION theory SPECTRUM analysis BEAM optics GAUSSIAN processes LIGHT modulators SOUND waves COLD gases