Fuzzy Logic Strikes Back: Fuzzy ODEs for Dynamic Modeling and Uncertainty Quantification

Traditionally, system identification (SysID) has emphasized point predictions and model accuracy, with limited exploration of prediction intervals (PIs) to assess uncertainty. Type-2 (T2) fuzzy logic systems (FLSs) are well-suited for uncertainty quantification (UQ), yet few methods have extended their application to learning temporal dynamics and generating prediction intervals (PIs) for SysID tasks. This article introduces fuzzy ordinary differential equations (FODEs), a novel network integrating the UQ capabilities of T2-FLSs with the representational strength of neural ordinary differential equations (NODEs) for system identification (SysID). Our model, FODE, leverages the nonlinearity representation capabilities of FLSs to capture system dynamics while its T2 variant is capable of producing PIs alongside point predictions. This article presents in-depth the mathematical foundations of Type-1, Interval T2, and General T2 (GT2) FODEs. We demonstrate that IT2-FODE and GT2-FODE effectively generate interval and granular uncertainty estimates, respectively. For training FODE, we present a deep-learning framework and introduce a modified Relaxed Quantile Regression loss to train T2-FODEs effectively for time-series data, incorporating both position-based and velocity-based components to enhance PI learning. Through extensive benchmarking across different SysID problems, we demonstrate FODE's superior performance in both accuracy and high-quality PI generation, outperforming NODEs, particularly with General T2 FODEs. This work highlights FODEs as a solution for SysID problems with inherent UQ, offering a more comprehensive approach to understanding and predicting system behavior.