SIAM Journal on Scientific Computing最新文献_第9页

An Entropy Stable Essentially Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Conservation Laws 双曲守恒定律的熵稳定基本无振荡非连续伽勒金方法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-04-01 DOI: 10.1137/22m1524151

Yong Liu, Jianfang Lu, Chi-Wang Shu

{"title":"An Entropy Stable Essentially Oscillation-Free Discontinuous Galerkin Method for Hyperbolic Conservation Laws","authors":"Yong Liu, Jianfang Lu, Chi-Wang Shu","doi":"10.1137/22m1524151","DOIUrl":"https://doi.org/10.1137/22m1524151","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1132-A1159, April 2024. Abstract. Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [T. Chen and C.-W. Shu, J. Comput. Phys., 345 (2017), pp. 427–461; J. Chan, J. Comput. Phys., 362 (2018), pp. 346–374; T. Chen and C.-W. Shu, CSIAM Trans. Appl. Math., 1 (2020), pp. 1–52] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299–1324; Y. Liu, J. Lu, and C.-W. Shu, SIAM J. Sci. Comput., 44 (2022), pp. A230–A259], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"53 97 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566135","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

Dimensions of Exactly Divergence-Free Finite Element Spaces in 3D 三维精确无发散有限元空间的尺寸

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-04-01 DOI: 10.1137/22m1544579

L. Ridgway Scott, Tabea Tscherpel

{"title":"Dimensions of Exactly Divergence-Free Finite Element Spaces in 3D","authors":"L. Ridgway Scott, Tabea Tscherpel","doi":"10.1137/22m1544579","DOIUrl":"https://doi.org/10.1137/22m1544579","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1102-A1131, April 2024. Abstract. We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in three dimensions with exact divergence constraints. More precisely, we compare the standard Scott–Vogelius elements of higher polynomial degree and low-order methods on split meshes, the Alfeld and the Worsey–Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of only a few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in the Stokes complex for finite element methods on the Worsey–Farin split. 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://git-ce.rwth-aachen.de/pub/meshquantities/ and in the supplementary materials (meshquantities-main.zip [3.13KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"85 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566348","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 Incremental Tensor Train Decomposition Algorithm 增量张量列车分解算法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-26 DOI: 10.1137/22m1537734

Doruk Aksoy, David J. Gorsich, Shravan Veerapaneni, Alex A. Gorodetsky

{"title":"An Incremental Tensor Train Decomposition Algorithm","authors":"Doruk Aksoy, David J. Gorsich, Shravan Veerapaneni, Alex A. Gorodetsky","doi":"10.1137/22m1537734","DOIUrl":"https://doi.org/10.1137/22m1537734","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1047-A1075, April 2024. Abstract. We present a new algorithm for incrementally updating the tensor train decomposition of a stream of tensor data. This new algorithm, called the tensor train incremental core expansion (TT-ICE), improves upon the current state-of-the-art algorithms for compressing in tensor train format by developing a new adaptive approach that incurs significantly slower rank growth and guarantees compression accuracy. This capability is achieved by limiting the number of new vectors appended to the TT-cores of an existing accumulation tensor after each data increment. These vectors represent directions orthogonal to the span of existing cores and are limited to those needed to represent a newly arrived tensor to a target accuracy. We provide two versions of the algorithm: TT-ICE and TT-ICE accelerated with heuristics (TT-ICE[math]). We provide a proof of correctness for TT-ICE and empirically demonstrate the performance of the algorithms in compressing large-scale video and scientific simulation datasets. Compared to existing approaches that also use rank adaptation, TT-ICE[math] achieves [math] higher compression and up to [math] reduction in computational time. 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/dorukaks/TT-ICE as well as in the accompanying supplementary material.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"33 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312135","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

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems 利用汉密尔顿-雅可比多项式来解决某些科学机器学习问题

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-15 DOI: 10.1137/23m1561397

Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis

{"title":"Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems","authors":"Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis","doi":"10.1137/23m1561397","DOIUrl":"https://doi.org/10.1137/23m1561397","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page C216-C248, April 2024. Abstract. Hamilton–Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multitime case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multitime Hopf formula, which corresponds to a representation of the solution to certain multitime HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multitime HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the linear quadratic regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Finally, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, posttraining calibration, transfer learning, and sparse dynamics identification.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"22 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140152231","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

On the Convergence of Monolithic Multigrid for Implicit Runge–Kutta Time Stepping of Finite Element Problems 论有限元问题隐式 Runge-Kutta 时间步进的单片多网格收敛性

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-13 DOI: 10.1137/23m1569344

Robert C. Kirby

{"title":"On the Convergence of Monolithic Multigrid for Implicit Runge–Kutta Time Stepping of Finite Element Problems","authors":"Robert C. Kirby","doi":"10.1137/23m1569344","DOIUrl":"https://doi.org/10.1137/23m1569344","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. Abstract. Finite element discretizations of time-dependent problems also require effective time-stepping schemes. While implicit Runge–Kutta methods provide favorable accuracy and stability properties, they give rise to large and complicated systems of equations to solve for each time step. These algebraic systems couple all Runge–Kutta stages together, giving a much larger system than for single-stage methods. We consider an approach to these systems based on monolithic smoothing. If stage-coupled smoothers possess a certain kind of structure, then the question of convergence of a two-grid or multigrid iteration reduces to convergence of a related strategy for a single-stage system with a complex-valued time step. In addition to providing a general theoretical approach to the convergence of monolithic multigrid methods, several numerical examples are given to illustrate the theory and show how higher-order Runge–Kutta methods can be made effective in practice. 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/rckirby/CodeForMMGPaper as well as in the supplemental material.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"112 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127839","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

Sparse Recovery of Elliptic Solvers from Matrix-Vector Products 从矩阵矢量乘积稀疏恢复椭圆求解器

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-12 DOI: 10.1137/22m154226x

Florian Schäfer, Houman Owhadi

{"title":"Sparse Recovery of Elliptic Solvers from Matrix-Vector Products","authors":"Florian Schäfer, Houman Owhadi","doi":"10.1137/22m154226x","DOIUrl":"https://doi.org/10.1137/22m154226x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A998-A1025, April 2024. Abstract. In this work, we show that solvers of elliptic boundary value problems in [math] dimensions can be approximated to accuracy [math] from only [math] matrix-vector products with carefully chosen vectors (right-hand sides). The solver is only accessed as a black box, and the underlying operator may be unknown and of an arbitrarily high order. Our algorithm (1) has complexity [math] and represents the solution operator as a sparse Cholesky factorization with [math] nonzero entries, (2) allows for embarrassingly parallel evaluation of the solution operator and the computation of its log-determinant, (3) allows for [math] complexity computation of individual entries of the matrix representation of the solver that, in turn, enables its recompression to an [math] complexity representation. As a byproduct, our compression scheme produces a homogenized solution operator with near-optimal approximation accuracy. By polynomial approximation, we can also approximate the continuous Green’s function (in operator and Hilbert–Schmidt norm) to accuracy [math] from [math] solutions of the PDE. We include rigorous proofs of these results. To the best of our knowledge, our algorithm achieves the best known trade-off between accuracy [math] and the number of required matrix-vector products. 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/f-t-s/sparse_recovery_of_elliptic_solution_operators_from_matrix-vector_products and in the supplementary materials (CompressingSolvers.jl-main.zip [2.50MB], sparse_recovery_of_elliptic_solution_operators_from_matrix-vector_products-main.zip [54.8KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"48 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117016","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

High-Order Mass- and Energy-Conserving Methods for the Nonlinear Schrödinger Equation 非线性薛定谔方程的高阶质量和能量守恒方法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-12 DOI: 10.1137/22m152178x

Genming Bai, Jiashun Hu, Buyang Li

{"title":"High-Order Mass- and Energy-Conserving Methods for the Nonlinear Schrödinger Equation","authors":"Genming Bai, Jiashun Hu, Buyang Li","doi":"10.1137/22m152178x","DOIUrl":"https://doi.org/10.1137/22m152178x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1026-A1046, April 2024. Abstract. A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schrödinger equation based on Gauss collocation in time and finite element discretization in space, by introducing a mass- and energy-correction post-process at every time level. The existence, uniqueness, and high-order convergence of the numerical solutions are proved. In particular, the error of the numerical solution is [math] in the [math] norm after incorporating the accumulation errors arising from the post-processing correction procedure at all time levels, where [math] and [math] denote the degrees of finite elements in time and space, respectively, which can be arbitrarily large. Several numerical examples are provided to illustrate the performance of the proposed new method, including the conservation of mass and energy and the high-order convergence in simulating solitons and bi-solitons. 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/jiashhu/ME-Conserved-NLS and in the supplementary materials (ME-Conserved-NLS-main-2-Reproducibility-badge.zip [14.4MB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"5 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117073","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 Numerical Method for the Stability Analysis of Linear Age-Structured Models with Nonlocal Diffusion 非局部扩散线性年龄结构模型稳定性分析的数值方法

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-11 DOI: 10.1137/23m1568971

Dimitri Breda, Simone De Reggi, Rossana Vermiglio

{"title":"A Numerical Method for the Stability Analysis of Linear Age-Structured Models with Nonlocal Diffusion","authors":"Dimitri Breda, Simone De Reggi, Rossana Vermiglio","doi":"10.1137/23m1568971","DOIUrl":"https://doi.org/10.1137/23m1568971","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A953-A973, April 2024. Abstract. We numerically address the stability analysis of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, models with nonlocal diffusion are more challenging since the associated semigroups have no regularizing properties in the spatial variable. Nevertheless, the asymptotic stability of the null equilibrium is determined by the spectrum of the infinitesimal generator associated with the semigroup. We propose a numerical method to approximate the leading part of this spectrum by first reformulating the problem via integration of the age-state and then by discretizing the generator combining a spectral projection in space with a pseudospectral collocation in age. A rigorous convergence analysis proving spectral accuracy is provided in the case of separable model coefficients. Results are confirmed experimentally and numerical tests are presented also for the more general instance. 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://gitlab.com/SIMONE.DEREGGI/agenonlocig and in the supplementary materials (DE_REGGI_M156897R_codes.zip [16KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"72 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105617","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

AAA Rational Approximation on a Continuum AAA 连续面上的合理近似值

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-11 DOI: 10.1137/23m1570508

Tobin A. Driscoll, Yuji Nakatsukasa, Lloyd N. Trefethen

{"title":"AAA Rational Approximation on a Continuum","authors":"Tobin A. Driscoll, Yuji Nakatsukasa, Lloyd N. Trefethen","doi":"10.1137/23m1570508","DOIUrl":"https://doi.org/10.1137/23m1570508","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A929-A952, April 2024. Abstract. AAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains; the latter two are equivalent by a Möbius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a AAA-Lawson option for improvement of a AAA approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain. 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://epubs.siam.org/doi/10.1137/23M1570508#supplementary-materials.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"16 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105694","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

Solving the Boltzmann Equation with a Neural Sparse Representation 用神经稀疏表示法求解玻尔兹曼方程

IF 3.1 2区数学

SIAM Journal on Scientific Computing Pub Date : 2024-03-11 DOI: 10.1137/23m1558227

Zhengyi Li, Yanli Wang, Hongsheng Liu, Zidong Wang, Bin Dong

{"title":"Solving the Boltzmann Equation with a Neural Sparse Representation","authors":"Zhengyi Li, Yanli Wang, Hongsheng Liu, Zidong Wang, Bin Dong","doi":"10.1137/23m1558227","DOIUrl":"https://doi.org/10.1137/23m1558227","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page C186-C215, April 2024. Abstract. We consider the neural sparse representation to solve the Boltzmann equation with BGK and quadratic collision models, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and physical space so as to avoid the discretization in space and time. Different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision models, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All of these will significantly improve the efficiency of the network when solving the Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multiscale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod tube problems and a 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"5 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105691","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