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.

翻译：降阶模型可加速高维动力学模拟，但当在线动力学偏离离线训练数据表征的工况区域时，其精度往往显著下降。我们基于增量奇异值分解（iSVD）构建了一种投影型自适应降阶框架，该框架通过间歇性调用全阶算子评估来获取修正快照，用于在线基函数更新。本文考虑的非侵入式降阶模型完全以基函数为参数化对象，因此每次基函数更新可自然传递至降阶算子与超降阶计算模块。凭借其演化中的奇异结构，iSVD能够编码已观测动力学的历史信息，在此意义上具有历史感知特性。我们在三个非线性问题（一维粘性Burgers方程、Sod激波管、以及刚性一维十物种旋转爆震发动机）上研究了该方法，且问题复杂度逐次递增。通过Burgers问题分析该方法，并将iSVD与替代基函数自适应规则进行比较，结果表明历史感知更新优于即时更新，且iSVD整体性能最为优越。Sod激波管与旋转爆震发动机案例证明，这些优势在更具挑战性的可压缩流动环境中依然存在。针对旋转爆震发动机问题，iSVD自适应降阶模型在预测精度与计算效率两方面均优于当前最先进的直接自适应降阶基准模型。成本分析表明，在线计算主要开销来自与全阶模型交互获取修正快照，而iSVD更新本身的计算代价可忽略不计。这些结果揭示iSVD是降阶子空间在线学习的有效机制，为开发预测跨度比初始训练窗口长数个量级的降阶模型指明了方向。