发表机构
University of Science and Technology of China; Brown University（中国科学技术大学; 布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对输运主导问题中降阶模型产生的非物理振荡，提出基于Gegenbauer多项式重构的后处理框架，有效消除伪振荡，将误差降低1-2个数量级。
AI 中文摘要
在本文中，我们为输运主导问题的数据驱动降阶模型（ROM）开发了一种基于物理的后处理技术。除了Kolmogorov n-width衰减缓慢之外，基于全局支撑基的ROM在逼近具有激波或尖锐梯度的解时，常常产生非物理振荡，这种现象类似于谱逼近中的Gibbs振荡。为了解决这个问题，我们引入了一个基于Gegenbauer多项式重构的后处理框架。关键思想是将ROM解重新投影到每个解析区间上的Gegenbauer多项式基上。Gegenbauer重构最初是为谱逼近开发的，它在有效抑制Gibbs振荡的同时实现了谱精度。我们将该技术扩展到数据驱动ROM，并考虑了三种代表性方法：本征正交分解（POD）-Galerkin ROM、算子推断（OpInf）和基于卷积自编码器（CAE）的非线性流形ROM。数值结果表明，所提出的后处理一致地消除了所有三种ROM的伪振荡，并显著提高了解质量。对于一维问题，一旦检测到不连续点，该方法实现简单。我们进一步开发了二维问题的实用扩展，即在每个坐标方向上进行逐行重构。大量数值实验表明，对于无粘输运问题，所提出的方法将误差降低多达1-2个数量级，并且在数值精度和间断的尖锐分辨率方面显著优于全变差正则化。
英文摘要
In this paper, we develop a physics-based post-processing technique for data-driven reduced-order models (ROMs) of transport-dominated problems. Besides the slow decay of the Kolmogorov n-width, ROMs based on globally supported bases often produce unphysical oscillations when approximating solutions with shocks or sharp gradients, a phenomenon analogous to Gibbs oscillations in spectral approximations. To address this issue, we introduce a post-processing framework based on Gegenbauer polynomial reconstruction. The key idea is to re-project the ROM solution onto a Gegenbauer polynomial basis over each interval of analyticity. Originally developed for spectral approximations, Gegenbauer reconstruction achieves spectral accuracy while effectively suppressing Gibbs oscillations. We extend this technique to data-driven ROMs and consider three representative approaches: Proper Orthogonal Decomposition (POD)-Galerkin ROM, Operator Inference (OpInf), and nonlinear manifold ROMs based on convolutional autoencoders (CAE). Numerical results show that the proposed post-processing consistently removes spurious oscillations and substantially improves solution quality for all three ROMs. For one-dimensional problems, the method is straightforward to implement once discontinuities are detected. We further develop a practical extension to two-dimensional problems using line-by-line reconstruction in each coordinate direction. Extensive numerical experiments demonstrate that the proposed method reduces errors by up to one or two orders of magnitude for inviscid transport problems and significantly outperforms total variation regularization in both numerical accuracy and the sharp resolution of discontinuities.
