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A qualitative study of the dynamic behavior for adaptive gradient algorithms

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Stanford University Program in Applied and Computational Mathematics Princeton University Program in Applied and Computational Mathematics Department of Mathematics Princeton University

The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

IMPROVED EFFICIENCY OF MULTILEVEL MONTE CARLO FOR STOCHASTIC PDE THROUGH STRONG PAIRWISE COUPLING

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arXiv 2021年

作者： Chada, Neil K. Hoel, Håkon Jasra, Ajay Zouraris, Georgios E. Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Mathematics for Uncertainty Quantification RWTH Aachen University Aachen Germany Department of Mathematics and Applied Mathematics Voutes Campus Crete HeraklionGR-700 13 Greece

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when MLMC is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the MLMC method with strong pairwise coupling that was developed and studied numerically on filtering problems in [Chernov et al., Numer. Math., 147 (2021), 71-125], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this *** Codes 65C05, 65N35, 60H35, 65C30 © 2021, CC BY.

关键词： Monte Carlo methods

Not all spacetime coordinates for general-relativistic ray tracing are created equal

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arXiv 2023年

作者： Bozzola, Gabriele Chan, Chi-Kwan Paschalidis, Vasileios Department of Astronomy University of Arizona TucsonAZ United States Data Science Institute University of Arizona TucsonAZ United States Program in Applied Mathematics University of Arizona TucsonAZ United States Department of Physics University of Arizona TucsonAZ United States

Models for the observational appearance of astrophysical black holes rely critically on accurate general-relativistic ray tracing and radiation transport to compute the intensity measured by a distant observer. In this paper, we illustrate how the choice of coordinates and initial conditions affect this process. In particular, we show that propagating rays from the camera to the source leads to different solutions if the spatial part of the momentum of the photon points towards the horizon or away from it. In doing this, we also show that coordinates that are well suited for numerical General-Relativistic MagnetoHydroDynamic (GRMHD) simulations are typically not optimal for generic ray tracing. We discuss the implications for black-hole images and show that radiation transport in optimal and non-optimal spacetime coordinates lead to the same images up to numerical errors and algorithmic choices. Copyright © 2023, The Authors. All rights reserved.

关键词： Ray tracing

Deep Learning based Fault Diagnosis in a Grid-Connected Photovoltaic Systems 19

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Deep Learning based Fault Diagnosis in a Grid-Connected Phot...

19th IEEE International Multi-Conference on Systems, Signals and Devices, SSD 2022

作者： Hichri, Amal Hajji, Mansour Mansouri, Majdi Bouzrara, Kais Nounou, Hazem Nounou, Mohamed Higher Institute of Applied Sciences and Technology of Kasserine Kasserine Tunisia Electrical and Computer Engineering Program Texas A&m University at Qatar Doha23874 Qatar Prince Sultan University Department of Mathematics and Sciences Riyadh11586 Saudi Arabia Signal Processing and Image National Engineering School of Monastir Research Laboratory of Automation 5019 Tunisia Chemical Engineering Program Texas A&m University at Qatar Doha23874 Qatar

ISBN: (数字)9781665471084

ISBN: (纸本)9781665471084

Photovoltaic (PV) system fault diagnosis is crucial because it helps PV system operators reduce energy and income losses. It also decreases the risk of fire and electric shock from PV system failures. Thus, the implementation of fault diagnosis tool is required. The most well-known techniques are machine learning models. Although these techniques provided well in terms of diagnosis and classification accuracy, they rely on manual feature extraction, which is time consuming, expensive, and requires diagnostic skill. Manual feature extraction is a problem that has to be addressed. This paper therefore presents a PV system fault diagnosis techniques that can automatically extract features from raw data in order to classify PV system faults. The implemented methods include long short-term memory (LSTM) bidirectional long short-term memory (BiLSTM) networks, which are deep learning based algorithms. Finally, the presented fault diagnosis approaches showed a good diagnosis performances when applied to an emulated GCPV system. © 2022 IEEE.

关键词： Failure analysis

Minuscule corrections to near-surface solar internal rotation using mode-coupling

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arXiv 2023年

作者： Das, Srijan Bharati Kashyap, Samarth G. Oktay, Deniz 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 Computer Science Princeton University PrincetonNJ United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow — differential rotation — is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency-splittings do not consider the effect of mode-coupling. This approximation worsens for high-angular-degree modes, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ` ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a non-linear inversion using an eigenvalue solver. Differences in inverted profiles for frequency splitting measurements from MDI and HMI are compared and discussed. We find that corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields. © 2023, CC BY.

关键词： Rotation

N-ACT: An Interpretable Deep Learning Model for Automatic Cell Type and Salient Gene Identification

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arXiv 2022年

作者： Ali Heydari, A. Davalos, Oscar A. Hoyer, Katrina K. Sindi, Suzanne S. Department of Applied Mathematics University of California United States Health Sciences Research Institute Quantitative and Systems Biology Graduate Program Department of Molecular and Cell Biology University of California Merced United States

Single-cell RNA sequencing (scRNAseq) is rapidly advancing our understanding of cellular composition within complex tissues and organisms. A major limitation in most scRNAseq analysis pipelines is the reliance on manual annotations to determine cell identities, which are time consuming, subjective, and require expertise. Given the surge in cell sequencing, supervised methods–especially deep learning models–have been developed for automatic cell type identification (ACTI), which achieve high accuracy and scalability. However, all existing deep learning frameworks for ACTI lack interpretability and are used as "black-box" models. We present N-ACT (Neural-Attention for Cell Type identification): the first-of-its-kind interpretable deep neural network for ACTI utilizing neural attention to detect salient genes for use in cell-types identification. We compare N-ACT to conventional annotation methods on two previously manually annotated data sets, demonstrating that N-ACT accurately identifies marker genes and cell types in an unsupervised manner, while performing comparably on multiple data sets to current state-of-the-art model in traditional supervised ACTI. © 2022, CC BY.

关键词： Genes

Flagellum Pumping Efficiency in Shear-Thinning Viscoelastic Fluids

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arXiv 2023年

作者： Barrett, Aaron Fogelson, Aaron L. Gregory Forest, M. Gruninger, Cole Lim, Sookkyung Griffith, Boyce E. Department of Mathematics University of Utah Salt Lake CityUT United States Departments of Mathematics and Biomedical Engineering University of Utah Salt Lake CityUT United States Department of Mathematics University of North Carolina Chapel HillNC United States Department of Applied Physical Sciences University of North Carolina Chapel HillNC United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Department of Mathematical Sciences University of Cincinnati CincinnatiOH United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g., sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the swimmer and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency, defined as a particular flow rate divided by the torque required to spin the motor, over this parameter space. Our findings indicate that motor efficiency (measured by the volumetric flow rate) can be boosted or degraded by relatively moderate or strong shear-thinning of the viscoelastic environment. Copyright © 2023, The Authors. All rights reserved.

关键词： Shear thinning

Generalization error of GAN from the discriminator's perspective

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arXiv 2021年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Beijing Institute of Big Data Research China

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for *** Codes 68T07, 62G07, 60-08 Copyright © 2021, The Authors. All rights reserved.

关键词： Probability distributions

Refined least squares for support recovery

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arXiv 2021年

作者： Lindenbaum, Ofir Steinerberger, Stefan Program in Applied Mathematics Yale University New HavenCT06511 United States Department of Mathematics University of Washington SeattleWA98195 United States

We study the problem of exact support recovery based on noisy observations and present Refined Least Squares (RLS). Given a set of noisy measurement y = Xθ∗ + ω, and X ∈ RN×D which is a (known) Gaussian matrix and ω ∈ RN is an (unknown) Gaussian noise vector, our goal is to recover the support of the (unknown) sparse vector θ∗ ∈ {-1, 0, 1}D. To recover the support of the θ∗ we use an average of multiple least squares solutions, each computed based on a subset of the full set of equations. The support is estimated by identifying the most significant coefficients of the average least squares solution. We demonstrate that in a wide variety of settings our method outperforms state-of-the-art support recovery algorithms. © 2021, CC BY.

关键词： Recovery