Fluid dynamics problems are characterized by being multidimensional and
nonlinear, causing the experiments and numerical simulations being complex,
time-consuming and monetarily expensive. In this sense, there is a need to find
new ways to obtain data in a more economical manner. Thus, in this work we
study the application of time series forecasting to fluid dynamics problems,
where the aim is to predict the flow dynamics using only past information. We
focus our study on models based on deep learning that do not require a high
amount of data for training, as this is the problem we are trying to address.
Specifically in this work we have tested three autoregressive models where two
of them are fully based on deep learning and the other one is a hybrid model
that combines modal decomposition with deep learning. We ask these models to
generate $200$ time-ahead predictions of two datasets coming from a numerical
simulation and experimental measurements, where the latter is characterized by
being turbulent. We show how the hybrid model generates more reliable
predictions in the experimental case, as it is physics-informed in the sense
that the modal decomposition extracts the physics in a way that allows us to
predict it.

本研究通过应用时间序列预测到流体动力学问题中，测试了基于深度学习的三种自回归模型，其中一个是与深度学习相结合的混合模型。研究发现，混合模型在具有湍流特性的实验数据中产生更可靠的预测，因为它借助模态分解从物理学角度提取了物理特征，从而实现了对流体动力学的预测。

探索利用模态分解的混合方法比全深度学习模型更有效的流体动力学预测

Exploring the efficacy of a hybrid approach with modal decomposition  over fully deep learning models for flow dynamics forecasting

Simple aerodynamic configurations under even modest conditions can exhibit
complex flows with a wide range of temporal and spatial features. It has become
common practice in the analysis of these flows to look for and extract
physically important features, or modes, as a first step in the analysis. This
step typically starts with a modal decomposition of an experimental or
numerical dataset of the flow field, or of an operator relevant to the system.
We describe herein some of the dominant techniques for accomplishing these
modal decompositions and analyses that have seen a surge of activity in recent
decades. For a non-expert, keeping track of recent developments can be
daunting, and the intent of this document is to provide an introduction to
modal analysis that is accessible to the larger fluid dynamics community. In
particular, we present a brief overview of several of the well-established
techniques and clearly lay the framework of these methods using familiar linear
algebra. The modal analysis techniques covered in this paper include the proper
orthogonal decomposition (POD), balanced proper orthogonal decomposition
(Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global
linear stability analysis, and resolvent analysis.

本研究论文介绍实现流体动力学中基于模态分解的分析方法，其中包括 Proper Orthogonal Decomposition（POD）和 Dynamic Mode Decomposition (DMD) 等关键词。