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arXiv 2019年

作者： Weinan, E. Ma, Chao Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert *** Codes 41A99, 49M99 Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

Machine learning from a continuous viewpoint

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arXiv 2019年

作者： Weinan, E. Ma, Chao Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, very much in the spirit of classical numerical analysis and statistical physics. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the shallow neural network model and the residual neural network model, can all be recovered as particular discretizations of different continuous formulations. We also present examples of new models, such as the ow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework. Copyright © 2019, The Authors. All rights reserved.

关键词： Machine learning

Analysis of the gradient descent algorithm for a deep neural network model with skip-connections

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arXiv 2019年

作者： Weinan, E. Ma, Chao Wang, Qingcan Wu, Lei Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Beijing Institute of Big Data Research

The behavior of the gradient descent (GD) algorithm is analyzed for a deep neural network model with skip-connections. It is proved that in the over-parametrized regime, for a suitable initialization, with high probability GD can find a global minimum exponentially fast. Generalization error estimates along the GD path are also established. As a consequence, it is shown that when the target function is in the reproducing kernel Hilbert space (RKHS) with a kernel defined by the initialization, there exist generalizable earlystopping solutions along the GD path. In addition, it is also shown that the GD path is uniformly close to the functions given by the related random feature model. Consequently, in this \implicit regularization" setting, the deep neural network model deteriorates to a random feature model. Our results hold for neural networks of any width larger than the input dimension. Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods

Cloaking the underlying long-range order of randomly perturbed lattices

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arXiv 2020年

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

Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying long-range order is then ‘cloaked’ in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations a, as measured by the hyperuniformity order metric Λ. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked or not as the perturbation strength increases, is manifestly captured by the τ order metric. Therefore, while the perturbation strength a may seem to be a natural choice for an order metric of perturbed lattices, the τ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least 106 particles) of disordered hyperuniform point patterns without Bragg peaks. Copyright © 2020, The Authors. All rights reserved.

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Interpretation of gravity satellite data to delineate structural features connected to geothermal resources at Bur Ni Geureudong geothermal field 3

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Interpretation of gravity satellite data to delineate struct...

3rd International Conference on Natural and Environmental Sciences 2019, ICONES 2019

作者： Putri, D.R. Nanda, M. Rizal, S. Idroes, R. Ismail, N. Doctoral Program of Mathematics and Applied Science Universitas Syiah Kuala Banda Aceh23111 Indonesia Marine Sciences Department Universitas Syiah Kuala Banda Aceh23111 Indonesia Department of Chemistry Faculty of Mathematics and Natural Sciences Universitas Syiah Kuala Kopelma Darussalam Banda Aceh23111 Indonesia Department of Pharmacy Faculty of Mathematics and Natural Sciences Universitas Syiah Kuala Kopelma Darussalam Banda Aceh23111 Indonesia Department of Physics Faculty of Mathematics and Natural Science Universitas Syiah Kuala Banda Aceh23111 Indonesia

The gravity method is broadly used in analyzing potential geothermal studies. The method can be used for determining potential areas, reservoir locations, and geological structure investigation. In this paper, satellite gravity data was applied for better understand the geological conditions connected with the geothermal system of Bur Ni Geureudong geothermal field, Bener Meriah District, Aceh, Indonesia. Open access Free Air anomaly data were provided by satellite geodesy information with 1 minute-grid. The data were reduced to obtain Bouguer anomaly distribution in the study area. Tilt derivative (TDR) technique was applied to Bouguer anomaly to enhance linear trends of geological structures. The aim of this analytical technique is able to clearly display faults correlated with geothermal manifestations around the Bur Ni Geureudong geothermal field. The complete Bouguer anomaly range of 20 mGal up to 170 mGal covers Bur Ni Geureudong volcano complex. A low Bouguer anomaly is located in all summits of volcanoes that are included in Bur Ni Geureudong Volcano Complex;those are summits of Bur Geureudong volcano, Bur Ni Telong Volcano, Pepanji Mountain, and Salah Nama Mountain. The low Bouguer anomaly is associated with the andesitic flow and volcanic ashes. The higher Bouguer Anomaly is reflected in high-density basement rock in the west of Geureudong Volcano. Tilt derivative (TDR) shows geological structures more detail, particularly, the tilt derivative (TDR) clearly detects two fault structures over the study area presented by a tilt value of zero. The results will be useful as basic information in exploration study of Bur Ni Geureudong geothermal field. © 2019 Published under licence by IOP Publishing Ltd.

关键词： Volcanoes

Local number fluctuations in hyperuniform and nonhyperuniform systems: Higher-order moments and distribution functions

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arXiv 2020年

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

The local number variance σ2(R) associated with a spherical sampling window of radius R enables a classification of many-particle systems in d-dimensional Euclidean space Rd according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness γ1(R), excess kurtosis γ2(R) and the corresponding probability distribution function P[N(R)] of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes and conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for γ1(R) and γ2(R) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on γ1(R), γ2(R) and P[N(R)] for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for γ1(R), γ2(R) and P[N(R)] are generated for each model. We also ascertain the proximity of P[N(R)] to the normal distribution via a novel Gaussian "distance" metric l2(R). Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes such that γ1(R) ∼ l2(R) ∼ R-(d+1)/2 and γ2(R) ∼ R-(d+1) for large R. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the "antihyperuniform" model studied here. We prove that one-dimensional hyperuniform systems of class I or any d-dimensional lattice cannot obey a CLT. Remarkably, we discovered that the gamma distribution provides a good approximation to P[N(R)] for

关键词： Higher order statistics

Modelling and forecasting daily average temperature in Bangkok, Thailand

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AIP Conference Proceedings 2024年第1期3123卷

作者： Kanyarat Thitiwatthanakan Manad Khamkong Ravi Lonkani Thanasak Mouktonglang Ph.D. Program in Applied Statistics Department of Statistics Faculty of Science Chiang Mai University 50200 Chiang Mai Thailand Department of Statistics Faculty of Science Chiang Mai University 50200 Chiang Mai Thailand Department of Finance Faculty of Business Administration Chiang Mai University 50200 Chiang Mai Thailand Department of Mathematics Faculty of Science Chiang Mai University 50200 Chiang Mai Thailand

In order to predict the occurrence of extreme climate events, forecasting and describing dynamic changes in temperature are essential. Understanding these events will allow action to be taken in order to minimize their associated effects. In this paper, we present a feasible model for estimating daily average temperatures in Bangkok, Thailand. The daily maximum and minimum temperature observations between 2006 and 2021 were collected from the Thai Meteorological department for the study. The Fourier series was then applied to the average daily temperature as the mean function, and then the auto-regressive integrated moving average, ARIMA (4,1,1), was used to predict the residuals that occurred from the mean function to describe the evolution of the temperature. Analysis of the root mean square error (RMSE) value from the models, which is 0.9536, revealed that the methods fitted the data quite well.

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Active learning of uniformly accurate interatomic potentials for materials simulation

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Physical Review Materials 2019年第2期3卷 023804-023804页

作者： Linfeng Zhang De-Ye Lin Han Wang Roberto Car Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA and Beijing Institute of Big Data Research Beijing 100871 People's Republic of China

An active learning procedure called deep potential generator (DP-GEN) is proposed for the construction of accurate and transferable machine learning-based models of the potential energy surface (PES) for the molecular modeling of materials. This procedure consists of three main components: exploration, generation of accurate reference data, and training. Application to the sample systems of Al, Mg, and Al-Mg alloys demonstrates that DP-GEN can produce uniformly accurate PES models with a minimal number of reference data.

关键词： Atomic & molecular structure Potential energy surfaces Alloys Amorphous materials Liquids Density functional theory First-principles calculations Molecular dynamics

Wave energy flux variability in the Caribbean sea

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Research Square

Research Square 2022年

作者： Orejarena-Rondón, Andrés F. Sayol, Juan-Manuel Hernández-Carrasco, Ismael Cáceres-Euse, Alejandro Restrepo, Juan C. Orfila, Alejandro Program of Marine and Coastal Geosciences INVEMAR Santa Marta Colombia Department of Applied Mathematics University of Alicante Sant Vicent del Raspeig Alicante03690 Spain Sistema de Observación Costero de les Illes Balears SOCIB Palma de Mallorca07007 Spain Esporles07190 Spain Mediterranean Institute of Oceanography MIO Université de Toulon Toulon83041 France Dpto. de Física y Geociencias Universidad del Norte Puerto Colombia081007 Colombia

Wave energy flux (WEF) is assessed in the Caribbean Sea from a 60-year (1958–2017) wave hindcast. We use a novel approach, based on neural networks, to identify coherent regions of similar WEF and their association with different climate patterns. This method allows for a better evaluation of the underlying dynamics behind seasonal and inter-annual WEF variability, including the effect induced by the latitudinal migration of the Intertropical Convergence Zone (ITCZ), and the influence of El Niño-Southern Oscillation events. Results show clear regional differences of the WEF variability likely due to both a clear regionalization of the WEF due to both the intensification and migration of the ITCZ. WEF exhibits a strong semiseasonal signal in areas of the continental shelf, with maximums in January and June, in agreement with the sea surface temperature and sea level pressure variability. At larger scales, WEF shows a significant correlation with the Oceanic Niño Index depicting positive values in the central and western basin and negative ones at the eastern side. © 2022, CC BY.

关键词： Surface temperature

Limiting regimes of turbulent horizontal convection. Part II: Large prandtl numbers

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arXiv 2020年

作者： Passaggia, Pierre-Yves Cohen, Nadia F. Scotti, Alberto White, Brian L. University of Orléans INSA-CVL PRISME EA 4229 Orléans45072 France Carolina Center for Interdisciplinary Applied Mathematics Dept. of Mathematics University of North Carolina Chapel HillNC27599 United States Department of Marine Sciences University of North Carolina Chapel HillNC27599 United States

Horizontal Convection (HC) at large Rayleigh and Prandtl numbers, is studied experimentally in a regime up to seven orders of magnitude larger in terms of Rayleigh numbers than previously achieved. To reach Rayleigh up to 1017, the horizontal density gradient is generated using differential solutal convection by a differential input of salt and fresh water controlled by diffusion in a novel experiment where the zero-net mass flux of water is ensured through permeable membranes. This setup allows for measuring accurately the Nusselt number in solutal convection by carefully controlling the amount of salt water exchanged through the membranes. Combined measurements of density and velocity across more than five orders of magnitude in Rayleigh numbers show that the flow transitions from the Beardsley & Festa [1, 39] regime to the Chiu-Webster et al.[3] regime and frames the present results within the scope of Shishkina et al.[38], and the companion paper[27] theory. In particular, we show that even for large Prandtl numbers, the circulation eventually clusters underneath the forcing horizontal boundary leaving a stratified core without motion. Finally, previous regime diagrams[12, 38] are extended combining the present results at high Prandtl numbers, the results at low Prandtl numbers of the companion paper, together with previous results from the literature. This work sets a new picture of the transition landscape of horizontal convection across six orders of magnitude in Prandtl number and sixteen orders of magnitude in Rayleigh numbers. Copyright © 2020, The Authors. All rights reserved.

关键词： Prandtl number