History-aware adaptive reduced-order models via incremental singular value decomposition

2026-05-27

AI Summary

This work addresses the significant accuracy degradation of traditional reduced-order models (ROMs) when online dynamics deviate from the offline training regime. To overcome this limitation, the authors propose an adaptive projection-based ROM framework leveraging incremental singular value decomposition (iSVD). The approach incorporates occasional full-order evaluations to acquire correction snapshots, enabling online updates of the reduced basis while simultaneously refining both the reduced operators and the hyper-reduction mechanism. A novel history-aware iSVD strategy is introduced to effectively preserve essential system evolution information, markedly outperforming instantaneous update methods. Numerical experiments on nonlinear dynamical systems—including the Burgers equation, Sod shock tube, and a rotating detonation engine—demonstrate that the proposed method surpasses existing adaptive ROM benchmarks in both predictive accuracy and computational efficiency, with negligible overhead from iSVD updates.

Abstract

Reduced-order models (ROMs) can accelerate high-dimensional dynamical simulations, but their accuracy often deteriorates when online dynamics leave the regime represented by offline training data. We develop a projection-based adaptive ROM framework based on incremental singular value decomposition (iSVD), in which occasional full-order operator evaluations provide correction snapshots for online basis updates. The intrusive ROMs considered here are fully parameterized by the basis, so each update naturally propagates to reduced operators and hyper-reduction machinery. Through its evolving singular structure, iSVD retains an encoded history of the observed dynamics and is history-aware in this sense. We study the method on three nonlinear problems of increasing complexity: the one-dimensional viscous Burgers equation, the Sod shock tube, and a stiff one-dimensional ten-species rotating detonation engine (RDE). The Burgers problem is used to analyze the method and compare iSVD with alternative basis adaptation rules, showing that history-aware updates outperform instantaneous updates and that iSVD gives the strongest overall performance. The Sod and RDE cases demonstrate that these advantages persist in more challenging compressible-flow settings. For the RDE problem, the iSVD adaptive ROM improves upon the current state-of-the-art Direct adaptive ROM baseline in both predictive accuracy and computational efficiency. A cost analysis shows that the dominant online cost comes from interacting with the full-order model to obtain correction snapshots, while the iSVD update itself is negligible. These results identify iSVD as an effective mechanism for online learning of reduced subspaces and suggest a path toward ROMs that remain predictive over horizons several orders of magnitude longer than their initial training window.

Problem

• reduced-order models
• online dynamics
• accuracy deterioration
• training data regime
• predictive capability

Innovation

• incremental singular value decomposition
• history-aware adaptation
• adaptive reduced-order models
• online basis update
• hyper-reduction