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

2026-04-16

AI Summary

This work proposes an interpretable modeling framework that integrates interval type-2 fuzzy logic systems (IT2-FLS) with ordinary differential equations (ODEs) to address the limited interpretability of uncertainty quantification in existing system identification methods. By employing an additive fuzzy ODE architecture, the model enables end-to-end training while simultaneously delivering high-accuracy point predictions and physically meaningful prediction intervals. A key innovation lies in constraining membership functions to activate only two adjacent rules, thereby substantially reducing rule overlap and significantly enhancing local transparency. Experimental results on standard system identification benchmarks demonstrate that the proposed approach achieves prediction accuracy comparable to that of fuzzy ODEs (FODE), while yielding superior prediction interval quality and markedly improved model interpretability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Uncertainty Quantification
• System Identification
• Interpretability
• Prediction Intervals
• Fuzzy ODE

Innovation

• Explainable AI
• Type-2 Fuzzy Logic
• Uncertainty Quantification
• System Identification
• Fuzzy ODE