基于采样低维马尔可夫动力学的预渐近恢复降阶模型
Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference
摘要：
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，所学习的模型正是传统上用模型降阶技术构造的降阶模型，所述模型降阶技术需要完全了解高维系统的控制方程和算子。因此，学习的模型保证继承了从传统的模型降阶得到的降阶模型的充分研究的性质。关键成分是一个新的数据采样方案，以获得重新投影的高维系统，对应于马尔可夫动力学在低维子空间的轨迹。精确恢复的降阶模型从这些重新投影的轨迹是保证前渐近在一定条件下的有限数量的数据和一个大类的系统与多项式非线性项。数值实验结果表明，该方法学习的低维模型与传统模型降阶到实际数值误差的降阶模型相匹配。数值结果进一步表明，低维模型拟合到重新投影的轨迹是预测的，即使在模型拟合到没有重新投影的轨迹是不准确和不稳定的情况下。
英文摘要：
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 model reduction techniques that require full knowledge of governing equations and operators of the high-dimensional systems. Thus, the learned models are guaranteed to inherit the well-studied properties of reduced models from traditional model reduction. 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 pre-asymptotically 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.
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