Post-Processing Reduced-Order Models for Transport-Dominated Problems by Gegenbauer Reconstruction

Jul 16, 2026

Develop numerical post-processing techniques for data-driven reduced-order models (ROMs) targeting transport-dominated problems. The work addresses slow Kolmogorov n-width decay and, more critically, unphysical oscillations caused by global reduced bases when solutions contain shocks or sharp gradients, linked to the Gibbs phenomenon. Gegenbauer polynomial reconstruction re-projects solutions on intervals of analyticity. Applied to POD-Galerkin ROMs, operator inference, and nonlinear manifold ROMs with convolutional auto-encoders, the method suppresses oscillations and reduces error by one to two orders for inviscid cases.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

Introduction

Motivation and problem setting

Connection to spectral methods and Gibbs-type oscillations

Existing ROM stabilization strategies

arXiv :2607 .01619v1 [math .NA] 2 Jul 2026 Post-Processing Reduced-Order Models for Transport-Dominated Problems by Gegenbauer Reconstruction Lei Yan ∗ Yan Jiang † Chi-Wang Shu ‡ Abstract In this paper, we develop numerical techniques for post-processing of data-driven reduced-order models (ROMs) for transport-dominated problems. For transport-dominated problems, we encounter not only the challenge of slow decay in the Kolmogorov nwidth, but also, due to the global nature of the reduced basis, the on-line computation of data-driven ROMs can result in unphysical oscillations when solving for solutions with shocks or sharp gradients. This is closely related to the Gibbs phenomenon, which is a well-known phenomenon in spectral approximations. This paper aims at addressing this issue using post-processing techniques from spectral methods by Gegenbauer polynomial reconstruction. The idea of Gegenbauer reconstruction is to re-project the solution in each interval of analyticity by Gegenbauer polynomial basis. It has been shown to offer spectral accuracy for spectral basis including Fourier and orthogonal polynomial expansions. We adopt this approach to ROM data. In particular, we consider three types of commonly used linear and nonlinear ROMs: Proper Orthogonal Decomposition (POD) based Galerkin ROM, Operator Inference (OpInf) and nonlinear manifold ROM based on convolutional auto-encoders (CAE) . We show that post-processing by Gegenbauer reconstruction is effective for all three ROMs in the sense that it eliminates unphysical oscillations for solutions with discontinuities. The Gegenbauer reconstruction can be easily applied in 1D once a discontinuity detector is used. We further propose a detailed reconstruction procedure for 2D problems combining line-by-line reconstruction with respect to each dimension. We demonstrate that Gegenbauer reconstruction can reduce the numerical error by one or two orders of magnitude for inviscid problems and the results significantly outperform standard method by total variation regularizationin numerical accuracy and sharp resolution of the discontinuity. Keywords—data-driven reduced-order models, proper orthogonal decomposition, operator inference, nonlinear manifold reduced-order models, Gegenbauer reconstruction, transport-dominated ∗ School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: lei [yan@mail.ustc.edu.cn](yan@mail.ustc.edu.cn). †School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. Email: [jiangy@ustc.edu.cn](jiangy@ustc.edu.cn. Research)[. Research](jiangy@ustc.edu.cn. Research) supported by NSFC grant 12271499. ‡Division of Applied Mathematics, Brown University, Providence, RI 02912, Email: chi-wang [shu@brown.edu](shu@brown.edu. Research)[. Research](shu@brown.edu. Research) supported by NSF grant DMS-2309249. 1 Introduction This paper introduces a post-processing technique for computing accurate solutions from datadriven reduced-order models (ROMs) for transport-dominated problems. It is well known that numerical solutions to conservation laws may develop shock and other complex structures, motivating decades of research on non-oscillatory numerical methods [13] . Those structures present great challenges for ROMs [30] . The goal of this paper is to design a numerical procedure to eliminate the spurious oscillations of solutions from ROMs for shocks. The main insight of our work comes from the fact that most commonly used data-driven ROMs are global methods, i.e. the basis functions or the representation of the solutions are defined on the whole domain. This characteristic establishes a close and fundamental connection between reduced-order models and spectral methods such as Chebyshev or Fourier expansions. On function approximation level, the spectral expansion of discontinuous functions gives rise to oscillatory behavior of the finite s

FAQs

What problem does the paper address in data-driven ROMs for transport-dominated equations?

It targets slow decay in Kolmogorov n-width and, especially, unphysical oscillations that appear when ROMs with global bases are used for solutions with shocks or sharp gradients.

How does Gegenbauer reconstruction remove spurious oscillations?

It re-projects the ROM solution on each interval of analyticity using a Gegenbauer polynomial basis, leveraging spectral-method post-processing ideas to suppress Gibbs-type behavior.

Which ROM types are tested, and what is the main outcome?

The approach is applied to POD-based Galerkin ROMs, operator inference (OpInf), and nonlinear manifold ROMs built with convolutional auto-encoders (CAE). Post-processing eliminates unphysical oscillations for discontinuous solutions and improves accuracy.

Jul 16, 2026

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