Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction
Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction
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DOI：
10.1016/j.jcp.2023.111913
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
Aleksandr Katrutsa;S. Utyuzhnikov;I. Oseledets
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文献类型：
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作者：
Aleksandr Katrutsa;S. Utyuzhnikov;I. Oseledets
动态模式分解已被证明是一种非常有效的技术来研究动态数据。这完全是一种数据驱动的方法，它从通常应该从测量中采样的数据快照中提取所有必要的信息。如果由于某些较小尺度的维度缺失或未测量而导致现有数据不完整，则这种方法的应用就会出现问题。这种设置经常发生在建模复杂的动态系统，如电网，特别是与降阶建模。考虑到未解决的变量的影响，基于Mori-Zwanzig形式主义的最优预测方法可以应用于获得最期望的预测在现有的不确定性。这有效地导致了一个时间预测模型的开发，该模型考虑了缺失数据的影响。在本文中，我们提供了一个详细的推导所考虑的方法从刘维尔方程，并完成它的优化问题，定义相应的观测数据的最佳过渡算子。与现有的方法相比，我们考虑一阶近似的Mori-Zwanzig分解，状态相应的优化问题，并解决它与基于梯度的优化方法。通过自动微分技术精确计算得到的目标函数的梯度。数值实验表明，所考虑的方法实际上给出了相同的动力学准确的Mori-Zwanzig分解，但计算量较小。
The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables, the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.