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This paper presents a unified mathematical framework for modeling dynamical systems that exhibit structured flows, such as source/sink behavior, cyclic dynamics, and topology-constrained transport. The authors build upon the Helmholtz-Hodge decomposition and Graph Vector Field framework to propose a hierarchy of modeling approaches with varying complexity, from simple parametric models to full graph-based representations on simplicial complexes. They introduce a systematic verification and validation methodology using ablation studies to identify which mechanisms (gradient, curl, harmonic, or topological) dominate observed dynamics.
Why it matters
This framework enables researchers to select appropriate models for complex systems based on available data and computational resources, while maintaining interpretability. The systematic approach for identifying dominant mechanisms could improve modeling across diverse fields including fluid dynamics, biological networks, climate systems, and engineered networks.
arXiv:2605.18250v3 Announce Type: replace
Abstract: 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.