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

arXiv: https://arxiv.org/abs/2605.28684 · HTML: https://mlsanity.com/paper/2605.28684/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 68 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 10 of 10)

- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [The Noisy Power Method: A Meta Algorithm with Applications](https://mlsanity.com/paper/1311.2495/) (2013)
- [First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate](https://mlsanity.com/paper/1607.07837/) (2016)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Lookahead data-gathering strategies for online adaptive model reduction of transport-dominated problems](https://mlsanity.com/paper/2307.14874/) (2023)
- [A globally convergent method to accelerate topology optimization using on-the-fly model reduction](https://mlsanity.com/paper/2004.05756/) (2020)
- [On GROUSE and Incremental SVD](https://mlsanity.com/paper/1307.5494/) (2013)
- [Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges](https://mlsanity.com/paper/2602.11378/) (2026)

## Built on by (most cited first, 1 of 1)

- [In-context learning from self-generated trajectories for adaptive model reduction](https://mlsanity.com/paper/2607.02937/) (2026)

## Abstract (arXiv)

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.