This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.

通过纯数据驱动的工作流程，构建了一套用于分布式动力系统的简化模型（ROMs）；所采用的ROMs是由近似惯性流形（AIMs）理论启发，并以此为模板；应用机器学习工具可以避免需要准确的截断Galerkin投影和推导闭合修正的需求；并探讨了通过自动编码器和扩散映射这类流形学习技术发现合适的潜在变量集并进行可解释性测试的方法；该方法可以用理论的（Fourier系数）、线性数据驱动的（POD模态）和/或非线性数据驱动的（扩散映射）坐标表示ROMs；同时描述了黑盒模型和（基于理论和数据纠正的）灰盒模型；灰盒模型是在截断Galerkin投影无法后处理的情况下必要的；文章使用Chafee-Infante反应扩散和Kuramoto-Sivashinsky耗散偏微分方程来举例并成功测试了整个框架。

非线性降维昔日与今日：机器学习时代耗散偏微分方程目标