arXiv: Numerical Analysis最新文献

Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints (Survey) 带点态约束的椭圆型分布最优控制问题的有限元方法(综述)

arXiv: Numerical Analysis Pub Date : 2020-08-18 DOI: 10.1007/978-3-030-42687-3_1

S. C. Brenner

引用次数: 5

A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions 求解不同边界条件椭圆型问题的深Galerkin法与深Ritz法的比较研究

arXiv: Numerical Analysis Pub Date : 2020-05-10 DOI: 10.4208/cmr.2020-0051

Jingrun Chen, Rui Du, Keke Wu

{"title":"A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions","authors":"Jingrun Chen, Rui Du, Keke Wu","doi":"10.4208/cmr.2020-0051","DOIUrl":"https://doi.org/10.4208/cmr.2020-0051","url":null,"abstract":"Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.","PeriodicalId":283112,"journal":{"name":"arXiv: Numerical Analysis","volume":"26 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124487157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 40

On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs 线性二阶椭圆型和抛物型偏微分方程的物理信息神经网络的收敛性

arXiv: Numerical Analysis Pub Date : 2020-04-03 DOI: 10.4208/cicp.oa-2020-0193

Yeonjong Shin, J. Darbon, G. Karniadakis

{"title":"On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs","authors":"Yeonjong Shin, J. Darbon, G. Karniadakis","doi":"10.4208/cicp.oa-2020-0193","DOIUrl":"https://doi.org/10.4208/cicp.oa-2020-0193","url":null,"abstract":"Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. \u0000As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.","PeriodicalId":283112,"journal":{"name":"arXiv: Numerical Analysis","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134452512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 165

A novel iterative penalty method to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamics problems 求解流体动力学问题有限体积POD-Galerkin降阶模型边界条件的迭代惩罚方法

arXiv: Numerical Analysis Pub Date : 2019-12-02 DOI: 10.4208/CICP.OA-2020-0059

S. Star, G. Stabile, F. Belloni, G. Rozza, J. Degroote

{"title":"A novel iterative penalty method to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamics problems","authors":"S. Star, G. Stabile, F. Belloni, G. Rozza, J. Degroote","doi":"10.4208/CICP.OA-2020-0059","DOIUrl":"https://doi.org/10.4208/CICP.OA-2020-0059","url":null,"abstract":"A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the reduced order models are 270-308 times faster than the full order models for the lid driven cavity test case and 13-24 times for the Y-junction test case.","PeriodicalId":283112,"journal":{"name":"arXiv: Numerical Analysis","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128381231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Linear and nonlinear fractional elliptic problems 线性和非线性分数型椭圆问题

arXiv: Numerical Analysis Pub Date : 2019-10-17 DOI: 10.1090/conm/754/15145

Juan Pablo Borthagaray, Wenbo Li, R. Nochetto

引用次数: 7

On the Generalized Method of Lines and its Proximal Explicit and Hyper-Finite Difference Approaches 关于直线的广义方法及其近显式和超有限差分方法

arXiv: Numerical Analysis Pub Date : 2019-04-28 DOI: 10.1201/9780429343315-30

F. Botelho

引用次数: 0

3. On negatively dependent sampling schemes, variance reduction, and probabilistic upper discrepancy bounds 3.负相关抽样方案，方差缩减和概率差异上限

arXiv: Numerical Analysis Pub Date : 2019-04-24 DOI: 10.1515/9783110652581-003

M. Gnewuch, Marcin Wnuk, N. Hebbinghaus