Sparse Identification of Truncation Errors
Sparse Identification of Truncation Errors
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DOI：
10.1016/j.jcp.2019.07.049
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
Stephan Thaler;Ludger Paehler;N. Adams
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文献类型：
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作者：
Stephan Thaler;Ludger Paehler;N. Adams
本文提出了一种数据驱动的方法来识别偏微分方程线性和非线性离散格式的空间和时间截断误差。出于截断误差的核心作用，例如在创建隐式大涡方案，我们介绍了稀疏识别截断误差（SITE）框架自动识别的条款修改后的微分方程从模拟数据。我们建立在数据驱动的发现和复杂系统的控制领域的最新进展，并将其与Warming，Hyett，Lerat和Peyret的修正微分方程分析的经典工作联合收割机相结合。我们增加了一个稀疏的回归有根的方法与适当的预处理例程，以帮助在个别修改后的微分方程项的识别。这种自定义算法管道的构建允许使用贝叶斯信息准则（BIC）衰减多重共线性效应以及自动调整稀疏回归超参数。作为概念的证明，我们限制分析有限差分格式，并留下其他数值方案开放供将来查询。测试用例包括线性平流方程的前向时间，向后空间离散，Burgers方程的MacCormack预测校正计划和Korteweg-de弗里斯方程的Zabusky和Kruska离散计划。基于变分研究，我们得到的离散化参数，预处理方法和稀疏回归算法的选择准则。结果显示高度准确的预测强调的承诺，SITE的分析和优化的离散化方案，修改后的微分方程的解析推导是不可行的。
This work presents a data-driven approach to the identification of spatial and temporal truncation errors for linear and nonlinear discretization schemes of Partial Differential Equations (PDEs). Motivated by the central role of truncation errors, for example in the creation of implicit Large Eddy schemes, we introduce theSparse Identification of Truncation Errors(SITE) framework to automatically identify the terms of the modified differential equation from simulation data. We build on recent advances in the field of data-driven discovery and control of complex systems and combine it with classical work on modified differential equation analysis of Warming, Hyett, Lerat and Peyret. We augment a sparse regression-rooted approach with appropriate preconditioning routines to aid in the identification of the individual modified differential equation terms. The construction of such a custom algorithm pipeline allows attenuating of multicollinearity effects as well as automatic tuning of the sparse regression hyperparameters using the Bayesian information criterion (BIC). As proof of concept, we constrain the analysis to finite difference schemes and leave other numerical schemes open for future inquiry. Test cases include the linear advection equation with a forward-time, backward-space discretization, the Burgers' equation with a MacCormack predictor-corrector scheme and the Korteweg-de Vries equation with a Zabusky and Kruska discretization scheme. Based on variation studies, we derive guidelines for the selection of discretization parameters, preconditioning approaches and sparse regression algorithms. The results showcase highly accurate predictions underlining the promise of SITE for the analysis and optimization of discretization schemes, where analytic derivation of modified differential equations is infeasible.