From agent-based models to Takagi–Sugeno linear differential equations (TS-LDEs): a fuzzy-logic bridge between discrete and continuous dynamics

Abstract

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–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.

Keywords Agent based modeling; Fuzzy logic; Linear differential equation; Takagi–Sugenu model