xFODE+: Explainable Type-2 Fuzzy Additive ODEs for Uncertainty Quantification

TL;DR

xFODE+ is an interpretable deep learning model for system identification that provides accurate point predictions and reliable uncertainty estimates with enhanced interpretability.

Contribution

It introduces a novel explainable Type-2 fuzzy additive ODE model that combines interpretability with uncertainty quantification in system identification.

Findings

xFODE+ matches FODE in prediction interval quality.

xFODE+ achieves comparable accuracy to existing models.

The model maintains interpretability through local rule constraints.

Abstract

Recent advances in Deep Learning (DL) have boosted data-driven System Identification (SysID), but reliable use requires Uncertainty Quantification (UQ) alongside accurate predictions. Although UQ-capable models such as Fuzzy ODE (FODE) can produce Prediction Intervals (PIs), they offer limited interpretability. We introduce Explainable Type-2 Fuzzy Additive ODEs for UQ (xFODE+), an interpretable SysID model which produces PIs alongside point predictions while retaining physically meaningful incremental states. xFODE+ implements each fuzzy additive model with Interval Type-2 Fuzzy Logic Systems (IT2-FLSs) and constraints membership functions to the activation of two neighboring rules, limiting overlap and keeping inference locally transparent. The type-reduced sets produced by the IT2-FLSs are aggregated to construct the state update together with the PIs. The model is trained in a DL framework via a composite loss that jointly optimizes prediction accuracy and PI quality. Results on benchmark SysID datasets show that xFODE+ matches FODE in PI quality and achieves comparable accuracy, while providing interpretability.