跨度内学习：利用自身预测调整降阶模型
In-context learning from self-generated trajectories for adaptive model reduction
- Department of Aerospace Engineering（航空航天工程系）
- Department of Electrical Engineering and Computer Science（电气工程与计算机科学系）
- Michigan Institute for Computational Discovery and Engineering, University of Michigan, Ann Arbor, MI, USA（美国密歇根大学安娜堡分校计算发现与工程研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究降阶模型在线动力学偏离训练数据时精度下降的问题，提出通过在当前降子空间内利用模型自身预测，经增量奇异值分解获得轨迹 informed 谱预处理器，使模型更有效吸收未来跨度外校正。
中文摘要 AI 辅助
降阶模型将高维动力学压缩为可快速评估的低维表示，但在线动力学漂移超出训练数据时会损失精度。自适应方法通过外部跨跨度信息更新子空间来解决此问题。我们发现当前降子空间中存在一个互补且以前未利用的跨跨度适应通道。通过将模型自身预测通过带遗忘的增量奇异值分解进行流处理，我们获得了一个轨迹 informed 谱预处理器，其中子空间不变，但基被重新加权并朝着动力学访问的模式重新对齐。这使模型能够更有效地吸收未来的跨跨度校正。我们在三维螺旋上展示了这种机制的各个方面，并在粘性 Burgers 和 Fisher-KPP 动力学上进行了验证。我们还讨论了如何将跨跨度学习视为上下文学习的动力学系统类似物。更广泛地说，跨跨度学习为计算科学提出了一个新原则，揭示了模型生成的轨迹包含比以前认识到的更多可用信息。
英文摘要
High-fidelity simulations of complex physical systems are often too expensive for repeated prediction, design, and control. Reduced-order models address this computational cost by compressing high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they often lose accuracy when online dynamics drift beyond the offline training data. Adaptive methods address this limitation by updating the reduced subspace online using external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. To exploit this channel, we continually update the reduced representation using the model's own predictions through an incremental singular-value decomposition with a forgetting factor. This produces a trajectory-informed spectral preconditioner in which the reduced subspace remains unchanged, while the basis is reweighted and realigned according to the directions visited by the evolving dynamics. This internal reorganization prepares the basis to absorb future out-of-span corrections more effectively, improving adaptation without requiring additional external information. We expose the mechanism through a three-dimensional spiral example and demonstrate its benefits on nonlinear partial differential equations, including viscous Burgers and Fisher--KPP dynamics. We also discuss how in-span learning can be interpreted as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.
