发表机构
The Pennsylvania State University; Texas Tech University（宾夕法尼亚州立大学; 德克萨斯理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出利用核岭回归（KRR）构建闭包建模框架，解决ODE/PDE差分方程闭包和动力学方程矩闭包两类问题，通过误差界分析和数值实验（Lorenz-63、Kuramoto-Sivashinsky及一维动力学方程）验证了其长期预测准确性和优于LSTM及全局模型的性能。
AI 中文摘要
我们开发了一个闭包建模框架，利用核岭回归（KRR）来识别动力系统中缺失的组成部分。该框架处理两类闭包问题：在常微分方程（ODE）和偏微分方程（PDE）设置中出现的差分方程闭包，以及由动力学方程中的矩闭包产生的代数闭包。对于第一类问题，我们在ODE设置中推导了一个误差界，该误差界量化了时间积分、未解析尺度的近似以及将未解析尺度效应耦合到已解析求解器所需的插值所贡献的误差。在Lorenz-63系统和Kuramoto-Sivashinsky方程上的数值实验表明，该方法能实现准确的长期预测，并且相对于基于LSTM的闭包模型有显著改进。对于第二类问题，我们通过将动力学通量与宏观通量之间的差异建模为已解析宏观变量的函数，来考虑一维动力学方程的矩闭包。我们比较了基于主成分分析（PCA）坐标的全局KRR模型与空间局部模型。虽然全局模型在单峰初始条件下表现良好，但在双峰初始条件下其准确性会下降。具有适当建模输入的空间局部模型提高了鲁棒性并实现了更高的预测准确性。
英文摘要
We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models with appropriate modeling inputs improve robustness and achieve higher predictive accuracy.
Comments31 pages, 8 figures
