Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference
Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference
复制标题
对低维马尔可夫动力学进行采样，通过算子推理从数据中轻松恢复简化模型
DOI：
10.1137/19m1292448
复制
发表时间：
2020
影响因子：
3.1
通讯作者：
Peherstorfer, Benjamin
中科院分区：
文献类型：
--
作者：
Peherstorfer, Benjamin
本文介绍了一种从高维黑箱动力系统的数据中学习低维模型的方法。新颖之处在于，学习到的模型正是传统上用经典的基于投影的模型约简技术构建的约简模型。因此，所提出的方法学习的模型保证具有从模型约简中已知的已被充分研究的简化模型的性质，而不需要完全了解控制方程，也不需要高维系统的算子。关键因素是一种新的数据采样方案，以获得高维系统的重投影轨迹，这些轨迹对应于低维子空间中的马尔可夫动力学。对于有限数量的数据和具有多项式非线性项的大系统，在一定条件下，保证了从这些重投影轨迹中精确恢复简化模型的预渐近性。数值结果表明，该方法得到的低维模型在实际应用中与传统模型约简到数值误差的约简模型相匹配。数值结果进一步表明，即使在没有重新投影的轨迹拟合模型不准确和不稳定的情况下，重新投影轨迹拟合的低维模型也具有预测能力。
This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models that are traditionally constructed with classical projection-based model reduction techniques. Thus, the proposed approach learns models that are guaranteed to have the well-studied properties of reduced models known from model reduction, without requiring full knowledge of the governing equations and without requiring the operators of the high-dimensional systems. The key ingredient is a new data sampling scheme to obtain re-projected trajectories of high-dimensional systems that correspond to Markovian dynamics in low-dimensional subspaces. The exact recovery of reduced models from these re-projected trajectories is guaranteed preasymptotically under certain conditions for finite amounts of data and for a large class of systems with polynomial nonlinear terms. Numerical results demonstrate that the low-dimensional models learned with the proposed approach match reduced models from traditional model reduction up to numerical errors in practice. The numerical results further indicate that low-dimensional models fitted to re-projected trajectories are predictive even in situations where models fitted to trajectories without re-projection are inaccurate and unstable.