检索结果-内蒙古大学图书馆

Hamiltonian neural networks for solving equations of motion

Physical Review E 2022年第6期105卷 065305-065305页

作者： Marios Mattheakis David Sondak Akshunna S. Dogra Pavlos Protopapas John A. Paulson School of Engineering and Applied Sciences Harvard University Cambridge Massachusetts 02138 USA Department of Mathematics Imperial College London London SW7 2AZ United Kingdom EPSRC CDT in Mathematics of Random Systems: Analysis Modelling and Algorithms London SW7 2AZ United Kingdom

There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the predictability of the network. Moreover, an error analysis is derived and states that the numerical errors depend on the overall network performance. The Hamiltonian network is then employed to solve the equations for the nonlinear oscillator and the chaotic Hénon-Heiles dynamical system. In both systems, a symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network to achieve the same order of the numerical error in the predicted phase space trajectories.

关键词： Classical mechanics

来源：评论

学校读者我要写书评

暂无评论

Error estimation and correction from within Neural Network Differential Equation solvers

arXiv

引用

arXiv 2020年

作者： Dogra, Akshunna S. Department of Mathematics Imperial College London United Kingdom EPSRC CDT in Mathematics of Random Systems: Analysis Modelling and Simulation

Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy interpret-ability of their models, etc. However, almost all NN DE solvers suffer from a fundamental limitation: they are trained using loss functions that depend only implicitly on the error associated with the estimate. As such, validation and error analysis of solution estimates requires knowledge of the true solution. Indeed, if the true solution is unknown, we are often reduced to simply hoping that a "low enough" loss implies "small enough" errors, since explicit relationships between the two are not available/well defined. In this work, we describe a general strategy for efficiently constructing error estimates and corrections for Neural Network Differential Equation solvers. Our methods do not require advance knowledge of the true solutions and obtain explicit relationships between loss functions and the error associated with solution estimates. In turn, these explicit relationships directly allow us to estimate and correct for the errors. Copyright © 2020, The Authors. All rights reserved.

关键词： Error analysis

来源：评论

学校读者我要写书评

暂无评论

SHAPER: Can You Hear the Shape of a Jet?

arXiv

引用

arXiv 2023年

作者： Ba, Demba Dogra, Akshunna S. Gambhir, Rikab Tasissa, Abiy Thaler, Jesse John A. Paulson School of Engineering and Applied Sciences Harvard University Western Ave CambridgeMA United States The NSF AI Institute for Artificial Intelligence and Fundamental Interactions United States Department of Mathematics Imperial College London Queen’s Gate London United Kingdom EPSRC CDT in Mathematics of Random Systems: Analysis Modelling and Simulation Oxford United Kingdom Center for Theoretical Physics Massachusetts Institute of Technology Massachusetts Ave CambridgeMA02139 United States Department of Mathematics Tufts University College Ave MedfordMA02155 United States

The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover’s Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (Shaper), which is a general framework for defining and computing shape-based observables. Shaper generalizes N-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the Shaper framework by performing empirical jet substructure studies using several examples of new shape-based observables. © 2023, CC BY.

关键词： High energy physics

来源：评论

学校读者我要写书评

暂无评论

Universality of Winning Tickets: A Renormalization Group Perspective

arXiv

引用

arXiv 2021年

作者： Redman, William T. Chen, Tianlong Wang, Zhangyang Dogra, Akshunna S. Interdepartmental Graduate Program in Dynamical Neuroscience University of California Santa Barbara United States Department of Electrical and Computer Engineering University of Texas at Austin United States Department of Mathematics Imperial College London United Kingdom EPSRC CDT in Mathematics of Random Systems: Analysis Modelling and Simulation United Kingdom

Foundational work on the Lottery Ticket Hypothesis has suggested an exciting corollary: winning tickets found in the context of one task can be transferred to similar tasks, possibly even across different architectures. This has generated broad interest, but methods to study this universality are lacking. We make use of renormalization group theory, a powerful tool from theoretical physics, to address this need. We find that iterative magnitude pruning, the principal algorithm used for discovering winning tickets, is a renormalization group scheme, and can be viewed as inducing a flow in parameter space. We demonstrate that ResNet-50 models with transferable winning tickets have flows with common properties, as would be expected from the theory. Similar observations are made for BERT models, with evidence that their flows are near fixed points. Additionally, we leverage our framework to study winning tickets transferred across ResNet architectures, observing that smaller models have flows with more uniform properties than larger models, complicating transfer between them. © 2021, CC BY.

关键词： Statistical mechanics

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：