Dynamic mode decomposition (DMD) has recently become a popular tool for the non-intrusive analysis of dynamical systems. Exploiting Proper Orthogonal Decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial basis evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots -- for different parameters and different time instants -- are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space the predicted reduced snapshots are then combined using regression techniques, thus enabling the possibility to approximate any untested parametric configuration in future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency and a fluid dynamics problem with nonlinear parameter dependency.

翻译：动态模态分解（DMD）近期已成为一种用于非侵入式动力系统分析的流行工具。通过利用本征正交分解（POD）作为降维技术，DMD能够将动力系统近似为一系列随时间线性演化的空间基函数的和，从而有助于深入理解物理现象并预测未来时刻的动态。本文针对参数化动力系统提出了一种DMD的扩展方法，重点关注参数化背景下对感兴趣输出的未来预测。首先，将所有快照——对应不同参数和不同时刻——投影到降阶空间；随后，采用DMD或其变体来近似未来时刻的降阶快照。利用降阶空间的低维特性，通过回归技术对预测得到的降阶快照进行组合，从而能够近似任意未经测试的参数配置在未来时刻的表现。本文详细阐述了该方法的算法核心；同时，我们展示并讨论了该算法的三个测试案例：具有线性参数依赖的简单动力系统、具有非线性参数依赖的热传导问题，以及具有非线性参数依赖的流体动力学问题。