SIAM Journal on Scientific Computing最新文献_第4页

Energetic Variational Neural Network Discretizations of Gradient Flows 梯度流的能量变异神经网络离散化

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-08-05 DOI: 10.1137/22m1529427

Ziqing Hu, Chun Liu, Yiwei Wang, Zhiliang Xu

{"title":"Energetic Variational Neural Network Discretizations of Gradient Flows","authors":"Ziqing Hu, Chun Liu, Yiwei Wang, Zhiliang Xu","doi":"10.1137/22m1529427","DOIUrl":"https://doi.org/10.1137/22m1529427","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2528-A2556, August 2024. Abstract. We present a structure-preserving Eulerian algorithm for solving [math]-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system’s free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0

An Alternating Flux Learning Method for Multidimensional Nonlinear Conservation Laws 多维非线性守恒定律的交替通量学习法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-08-01 DOI: 10.1137/23m1556605

Qing Li, Steinar Evje

{"title":"An Alternating Flux Learning Method for Multidimensional Nonlinear Conservation Laws","authors":"Qing Li, Steinar Evje","doi":"10.1137/23m1556605","DOIUrl":"https://doi.org/10.1137/23m1556605","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C421-C447, August 2024. Abstract. In a recent work [Q. Li and S. Evje, Netw. Heterog. Media, 18 (2023), pp. 48–79], it was explored how to identify the unknown flux function in a one-dimensional scalar conservation law. Key ingredients are symbolic neural networks to represent the candidate flux functions, entropy-satisfying numerical schemes, and a proper combination of initial data. The purpose of this work is to extend this methodology to a two-dimensional scalar conservation law ([math]) [math]. Straightforward extension of the method from the 1D to the 2D problem results in poor identification of the unknown [math] and [math]. Relying on ideas from joint and alternating equations training, a learning strategy is designed that enables accurate identification of the flux functions, even when 2D observations are sparse. It involves an alternating flux training approach where a first set of candidate flux functions obtained from joint training is improved through an alternating direction-dependent training strategy. Numerical investigations demonstrate that the method can effectively identify the true underlying flux functions [math] and [math] in the general case when they are nonconvex and unequal.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0

Graph Neural Reaction Diffusion Models 图形神经反应扩散模型

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-08-01 DOI: 10.1137/23m1576700

Moshe Eliasof, Eldad Haber, Eran Treister

引用次数: 0

A Semi-Implicit Fully Exactly Well-Balanced Relaxation Scheme for the Shallow Water System 浅水系统的半隐式完全精确均衡松弛方案

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-31 DOI: 10.1137/23m1621289

Celia Caballero-Cárdenas, Manuel Jesús Castro, Christophe Chalons, Tomás Morales de Luna, María Luz Muñoz-Ruiz

引用次数: 0

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks 用奇异性丰富物理信息神经网络解决多边形域中的泊松问题

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-30 DOI: 10.1137/23m1601195

Tianhao Hu, Bangti Jin, Zhi Zhou

{"title":"Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks","authors":"Tianhao Hu, Bangti Jin, Zhi Zhou","doi":"10.1137/23m1601195","DOIUrl":"https://doi.org/10.1137/23m1601195","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C369-C398, August 2024. Abstract. Physics-informed neural networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN, by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition, and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/hhjc-web/SEPINN.git and in the supplementary materials (M160119_SuppMat.pdf [399KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0

Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes 针对一类部分观测的片断确定性马尔可夫过程的多级粒子过滤器

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-26 DOI: 10.1137/23m1600505

Ajay Jasra, Kengo Kamatani, Mohamed Maama

引用次数: 0

Domain Decomposition Learning Methods for Solving Elliptic Problems 解决椭圆问题的领域分解学习方法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-23 DOI: 10.1137/22m1515392

Qi Sun, Xuejun Xu, Haotian Yi

{"title":"Domain Decomposition Learning Methods for Solving Elliptic Problems","authors":"Qi Sun, Xuejun Xu, Haotian Yi","doi":"10.1137/22m1515392","DOIUrl":"https://doi.org/10.1137/22m1515392","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2445-A2474, August 2024. Abstract. With the aid of hardware and software developments, there has been a surge of interest in solving PDEs by deep learning techniques, and the integration with domain decomposition strategies has recently attracted considerable attention due to its enhanced representation and parallelization capacity of the network solution. While there are already several works that substitute the numerical solver of overlapping Schwarz methods with the deep learning approach, the nonoverlapping counterpart has not been thoroughly studied yet because of the inevitable interface overfitting problem that would propagate the errors to neighboring subdomains and eventually hamper the convergence of outer iteration. In this work, a novel learning approach, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable the flux transmission across subregion interfaces with guaranteed accuracy, thereby allowing us to construct effective learning algorithms for realizing the more general nonoverlapping domain decomposition methods in the presence of overfitted interface conditions. Numerical experiments on a series of elliptic boundary value problems, including the regular and irregular interfaces, low and high dimensions, and smooth and high-contrast coefficients on multidomains, are carried out to validate the effectiveness of our proposed domain decomposition learning algorithms. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available\" as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/AI4SC-TJU or in the supplementary materials.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0

A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities 非线性变分不等式的全逼近方案多层次方法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-23 DOI: 10.1137/23m1594200

Ed Bueler, Patrick E. Farrell

{"title":"A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities","authors":"Ed Bueler, Patrick E. Farrell","doi":"10.1137/23m1594200","DOIUrl":"https://doi.org/10.1137/23m1594200","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2421-A2444, August 2024. Abstract. We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a joint extension of both the full approximation scheme multigrid technique for nonlinear partial differential equations, due to A. Brandt, and the constraint decomposition (CD) method introduced by X.-C. Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates. The full multigrid cycle version is an optimal solver. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://bitbucket.org/pefarrell/fascd/, where the software used to produce the results in section 8 is archived at tag v1.0, and at https://doi.org/10.5281/zenodo.10476845 or in the supplementary materials (pefarrell-fascd-6407e9f547d6.zip [225KB]). The authors used Firedrake master revision c5e939dde.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0

Computing [math]-Conforming Finite Element Approximations Without Having to Implement [math]-Elements 无需执行[数学]元素即可计算[数学]符合有限元近似值

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-22 DOI: 10.1137/23m1615486

Mark Ainsworth, Charles Parker

引用次数: 0

Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems 高频亥姆霍兹问题的超定位正交分解

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-07-18 DOI: 10.1137/21m1465950

Philip Freese, Moritz Hauck, Daniel Peterseim

{"title":"Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems","authors":"Philip Freese, Moritz Hauck, Daniel Peterseim","doi":"10.1137/21m1465950","DOIUrl":"https://doi.org/10.1137/21m1465950","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2377-A2397, August 2024. Abstract. We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber [math]. On a coarse mesh of width [math], the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width [math] and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter [math] is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition [math] with [math] denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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: 0