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

arXiv cs.LG / 4/17/2026

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Key Points

The paper proposes xFODE+, an interpretable Type-2 Fuzzy Additive ODE model for system identification that delivers both point predictions and prediction intervals for uncertainty quantification.
Unlike earlier fuzzy ODE approaches with limited interpretability, xFODE+ uses Interval Type-2 Fuzzy Logic Systems with constraints on rule activation to keep inference locally transparent and reduce unwanted overlap.
It aggregates type-reduced sets from the IT2-FLSs to compute state updates while simultaneously constructing prediction intervals.
Training is done in a deep learning framework using a composite loss that jointly improves prediction accuracy and the quality of the prediction intervals.
Experiments on benchmark system-identification datasets indicate xFODE+ matches FODE’s prediction-interval quality and achieves similar accuracy, with added interpretability benefits.

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.