Title: A Unified Framework for Structured Flow Modeling: From Continuous Fields to Data-Driven Representations

Abstract: A wide array of physical, engineered, and data-driven systems can be characterized by structured flows that integrate cyclic dynamics, source and sink behaviors, and transport processes constrained by topology. This study presents a cohesive perspective on these systems by linking continuous formulations rooted in Helmholtz-Hodge decomposition with discrete and data-centric representations. We examine the recently introduced Graph Vector Field (GVF) framework, which facilitates the breakdown of intricate dynamics into gradient, curl, and harmonic elements within simplicial complexes, thereby balancing interpretability with expressive capacity. Furthermore, we propose a tiered hierarchy of alternative modeling strategies—such as parametric conditional models, linear graph dynamical systems, and simplified Hodge representations—that prioritize computational efficiency and lower data demands at the expense of some expressive power. A central contribution of this paper is a cross-domain validation methodology that utilizes datasets from well-established physical systems to confirm model accuracy and evaluate robustness, irrespective of the specific application domain. This strategy allows for a systematic assessment of the balance between model complexity, interpretability, and predictive accuracy. The proposed framework fosters an iterative modeling process where highly expressive models serve as diagnostic instruments to pinpoint dominant mechanisms, thereby informing the development of simplified models suited to practical limitations. Ultimately, this work underscores the extensive utility of structured flow modeling and establishes a basis for the scalable, interpretable analysis of complex dynamical systems.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC