Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of physical, engineered, and data-driven systems. The objective of this work is to establish a unified perspective on such systems, to identify modeling approaches that balance expressivity, interpretability, computational complexity, and data requirements, and to investigate how highly expressive models can be used to uncover the dominant mechanisms underlying observed dynamics. Starting from the Helmholtz-Hodge decomposition of continuous vector fields, we review the recently proposed Graph Vector Field (GVF) framework and its discrete representation on simplicial complexes. We then introduce a hierarchy of alternative approaches, including parametric conditional models, linear graph dynamical systems, and reduced Hodge representations. Finally, we propose a verification and validation methodology based on benchmark datasets from well-understood physical systems and on systematic model-reduction and ablation studies. The resulting family of structured-flow models within a common framework, ranging from low-dimensional parametric representations to full GVF formulations, supports a diagnostic methodology in which gradient, curl, harmonic, and topological contributions are systematically assessed through ablation studies. This process enables the identification of dominant mechanisms underlying the observed dynamics and guides the construction of simplified models tailored to the available data and operational constraints. By separating structural verification, behavioral verification, and domain-specific validation, the proposed approach provides a foundation for scalable and interpretable analysis of complex dynamical systems across multiple application domains.

翻译：许多动力系统可以描述为结合源/汇行为、循环动力学和拓扑约束输运的结构化流。这些特征广泛存在于物理、工程和数据驱动系统中。本文旨在建立此类系统的统一视角，识别兼顾表达性、可解释性、计算复杂度和数据需求的建模方法，并探索如何利用高表达性模型揭示观测动力学背后的主导机制。从连续向量场的亥姆霍兹-霍奇分解出发，我们回顾了新近提出的图向量场框架及其在单纯复形上的离散表示。随后引入层次化替代方法，包括参数化条件模型、线性图动力系统和简化霍奇表示。最终提出基于已知物理系统基准数据集及系统性模型简化与消融研究的验证与确认方法论。由此在统一框架内形成的结构化流模型家族——从低维参数化表示到完整GVF公式——支持一种诊断方法，通过消融研究系统评估梯度、旋度、调和与拓扑贡献。该过程能够识别观测动力学背后的主导机制，并指导构建适配可用数据与操作约束的简化模型。通过分离结构验证、行为验证和特定领域确认，所提方法为跨多个应用领域的复杂动力系统可扩展且可解释的分析奠定了基础。