Online adaptive model reduction efficiently reduces numerical models of transport-dominated problems by updating reduced spaces over time, which leads to nonlinear approximations on latent manifolds that can achieve a faster error decay than classical linear model reduction methods that keep reduced spaces fixed. Critical for online adaptive model reduction is coupling the full and reduced model to judiciously gather data from the full model for adapting the reduced spaces so that accurate approximations of the evolving full-model solution fields can be maintained. In this work, we introduce lookahead data-gathering strategies that predict the next state of the full model for adapting reduced spaces towards dynamics that are likely to be seen in the immediate future. Numerical experiments demonstrate that the proposed lookahead strategies lead to accurate reduced models even for problems where previously introduced data-gathering strategies that look back in time fail to provide predictive models. The proposed lookahead strategies also improve the robustness and stability of online adaptive reduced models.

翻译：在线自适应模型降阶通过随时间更新降阶空间，高效缩减输运主导问题的数值模型，这导致潜在流形上的非线性近似，能够比保持降阶空间固定的经典线性模型降阶方法实现更快的误差衰减。在线自适应模型降阶的关键在于耦合全阶模型与降阶模型，以从全阶模型中审慎收集数据用于更新降阶空间，从而维持对演化全阶模型解场的精确近似。本文提出预判数据收集策略，通过预测全阶模型的下一状态，使降阶空间适应短期内可能出现的动力学行为。数值实验表明，即使对于以往回溯式数据收集策略无法提供预测模型的问题，所提出的预判策略仍能构建精确的降阶模型。同时，该策略还提升了在线自适应降阶模型的鲁棒性与稳定性。