This paper introduces the Takagi–Sugeno Latent Differential Equation (TS-LDE) framework as a gray-box modeling paradigm that bridges the gap between the microscopic interpretability of agent-based models (ABM) and the analytical tractability of continuous-time dynamical systems. Unlike traditional black-box simulations, TS-LDE explicitly captures latent system dynamics while preserving transparency in the rule-based structure, enabling both explanatory insight and predictive capability. To demonstrate the applicability of the proposed framework, we present four illustrative domains, climate dynamics (CO \(_2\) –temperature interaction), epidemic spreading (SIR), financial contagion, and water/agricultural resource management, as pedagogical demonstrations of the TS-LDE identification pipeline. Each case study serves as a step-by-step illustration of the modeling process: from ABM-inspired data generation, local parameter identification, and fuzzy rule construction, to TS-LDE simulation and sensitivity analysis. Our results show that the TS-LDE models reproduce the essential dynamics observed in their ABM counterparts while offering enhanced stability, smoother trajectories, and greater interpretability. This gray-box approach thus provides a structured and computationally efficient alternative to purely agent-based simulations, serving as a unifying surrogate framework for the analysis of complex systems across domains. The proposed methodology highlights how data-driven yet interpretable dynamical modeling can support deeper understanding, policy assessment, and pedagogical exploration of interconnected socio-environmental and financial systems.
