Acta Numerica最新文献_第4页

Asymptotic-preserving schemes for multiscale physical problems 多尺度物理问题的渐近保持格式

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-12-11 DOI: 10.1017/S0962492922000010

Shi Jin

引用次数: 25

ANU volume 30 Cover and Front matter 澳大利亚国立大学第30卷封面和封面问题

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/s096249292100009x

R. Altmann, P. Henning, D. Peterseim, P. Bartlett, A. Montanari, A. Rakhlin, Ronald A. DeVore, B. Hanin

引用次数: 0

Numerical homogenization beyond scale separation 超越尺度分离的数值均质化

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000015

R. Altmann, P. Henning, D. Peterseim

引用次数: 44

Optimal transportation, modelling and numerical simulation 最优运输，建模和数值模拟

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000040

J. Benamou

引用次数: 13

Tensors in computations 计算中的张量

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000076

Lek-Heng Lim

引用次数: 23

Learning physics-based models from data: perspectives from inverse problems and model reduction 从数据中学习基于物理的模型:从反问题和模型简化的角度

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000064

O. Ghattas, K. Willcox

{"title":"Learning physics-based models from data: perspectives from inverse problems and model reduction","authors":"O. Ghattas, K. Willcox","doi":"10.1017/S0962492921000064","DOIUrl":"https://doi.org/10.1017/S0962492921000064","url":null,"abstract":"This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"445 - 554"},"PeriodicalIF":14.2,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47260397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 69

Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation 无所畏惧:通过插补棱镜的深度学习的非凡数学现象

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/S0962492921000039

M. Belkin

引用次数: 116

ANU volume 30 Cover and Back matter 澳大利亚国立大学第30卷封面和封底

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-05-01 DOI: 10.1017/s0962492921000106

引用次数: 0

Modelling and computation of liquid crystals 液晶的建模与计算

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-04-06 DOI: 10.1017/S0962492921000088

Wen Wang, Lei Zhang, Pingwen Zhang

引用次数: 34

Deep learning: a statistical viewpoint 深度学习:统计学观点

IF 14.2 1区数学

Acta Numerica Pub Date : 2021-03-16 DOI: 10.1017/S0962492921000027

P. Bartlett, A. Montanari, A. Rakhlin

{"title":"Deep learning: a statistical viewpoint","authors":"P. Bartlett, A. Montanari, A. Rakhlin","doi":"10.1017/S0962492921000027","DOIUrl":"https://doi.org/10.1017/S0962492921000027","url":null,"abstract":"The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"87 - 201"},"PeriodicalIF":14.2,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492921000027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41948779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 177