A Unified Framework for Structured Flow Modeling: From Continuous Fields to Data-Driven Representations
For researchers modeling complex dynamical systems, this provides a unified framework for interpretable and scalable analysis, but it is primarily a review and methodological synthesis without new empirical results.
This paper unifies continuous and discrete representations of structured flows using the Helmholtz-Hodge decomposition and Graph Vector Field framework, enabling decomposition into gradient, curl, and harmonic components. It introduces a hierarchy of models trading expressivity for tractability and validates them across domains, supporting iterative model simplification.
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 domains, including physical, engineered, and data-driven systems. This work provides a unified perspective on such systems by connecting continuous formulations based on the Helmholtz-Hodge decomposition with discrete and data-driven representations. We review the recently proposed Graph Vector Field (GVF) framework, which enables a decomposition of complex dynamics into gradient, curl, and harmonic components on simplicial complexes, offering both expressivity and interpretability. We then introduce a hierarchy of alternative modeling approaches, including parametric conditional models, linear graph dynamical systems, and reduced Hodge representations, which trade expressive power for computational tractability and reduced data requirements. A key contribution of this work is a cross-domain validation strategy that leverages datasets from well-understood physical systems to verify model correctness and assess robustness independently of the target application domain. This approach enables a systematic evaluation of the trade-offs between model complexity, interpretability, and predictive performance. The resulting framework supports an iterative modeling methodology in which highly expressive models are used as diagnostic tools to identify dominant mechanisms, guiding the construction of simplified models tailored to practical constraints. This work highlights the broad applicability of structured flow modeling and provides a foundation for scalable and interpretable analysis of complex dynamical systems.