Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

Qusai Khaled; Uzay Kaymak; Laura Genga

cs.LGFeb 22, 2026

Alternating Bi-Objective Optimization for Explainable Neuro-Fuzzy Systems

Authors: Qusai Khaled, Uzay Kaymak, Laura Genga

Organizations: Jheronimus Academy of Data Science Eindhoven University of Technology Eindhoven, The Netherlands · School of Industrial Engineering Eindhoven University of Technology Eindhoven, The Netherlands

Abstract

Fuzzy systems show strong potential in explainable AI due to their rule-based architecture and linguistic variables. Existing approaches navigate the accuracy-explainability trade-off either through evolutionary multi-objective optimization (MOO), which is computationally expensive, or gradient-based scalarization, which cannot recover non-convex Pareto regions. We propose X-ANFIS, an alternating bi-objective gradient-based optimization scheme for explainable adaptive neuro-fuzzy inference systems. Cauchy membership functions are used for stable training under semantically controlled initializations, and a differentiable explainability objective is introduced and decoupled from the performance objective through alternating gradient passes. Validated in approximately 5,000 experiments on nine UCI regression datasets, X-ANFIS consistently achieves target distinguishability while maintaining competitive predictive accuracy, recovering solutions beyond the convex hull of the MOO Pareto front.

Figures & tables

Type	Considerations	Measures / Constraints
Structural interpretability	Number of rules, conditions, features, fuzzy sets, rule length	Pruning, rule merging, simplification, feature selection, rule base compression
Semantic interpretability	Distinguishability, coverage, normality, completeness, consistency, linguistic coherence	Parameter tuning, membership function shaping, overlap control, constraint enforcement, merging

TABLE I : Interpretability Types with Key Considerations and Measures

	Parameter	Initialization	Forward Pass	Backward Pass	X-Pass
Antecedents	$\mu$	FCM	Fixed ∗∗	GD	GD
	$\sigma$	FCM	Fixed	GD	Fixed
Consequents	$w$	N/A*	LSE+ $L_{2}$	Fixed	Fixed

TABLE II : Parameter Update Protocol for Alternating Bi-objective Optimization of X-ANFIS

No	Model	MF Type	R	Experiments
Initialisation Experiments
1	ANFIS	Gaussian	30	6
2	ANFIS	Cauchy	30	6
Pareto Experiments
3	ANFIS	Gaussian	5	9
4	MO-ANFIS	Cauchy	5	9x500

TABLE III : Summary of Experimental Design

#	Dataset Name	Samples	Features	Prediction Target
1	Airfoil Self-Noise	1,503	5	Sound pressure
2	Concrete Compressive Strength	1,030	8	Concrete strength
3	Energy Efficiency	768	8	Heating and cooling
4	Yacht Hydrodynamics	308	6	Sailing resistance
5	Bike Sharing (Day)	731	13	Daily count
6	Bike Sharing (Hour)	17,379	16	Hourly count

TABLE IV : Summary of Regression Real-world Datasets Used in Experiments Obtained from UCI Machine Learning Repository

Fig. 1 : Effect of MF initialization on ANFIS parameter update trajectories for the Combined Cycle Power Plant dataset, for humidity and vacuum features. Columns correspond to different initialized spread values $\sigma$ / $\gamma$ values; left three columns are Gaussian MFs, right three are Cauchy MFs.

MF Type	Initialized Std/Scale	MSE	RMSE	MAE	R²
Gaussian	1	0.0032	0.0563	0.0446	0.9377
	0.5	0.0030	0.0546	0.0428	0.9413
	0.25	0.0029	0.0539	0.0421	0.9428
	0.125	0.0038	0.0614	0.0487	0.9260
	0.0625	0.2513	0.5013	0.4477	-3.9394
	0.03125	0.2490	0.4990	0.4448	-3.8937

TABLE V : Test performance of Gaussian and Cauchy ANFIS models under different initialization scales

Fig. 2 : Training results of 500 models depicted as R² versus mean distinguishability for ANFIS, X-ANFIS and MO-ANFIS models. MO-ANFIS Pareto front shown in red. Classic ANFIS shown in pink. X-ANFIS shown in green.

Dataset	ANFIS				MO-ANFIS		X-ANFIS
	$\mathbf{R^{2}}$	$\mathbf{D}$	$\mathbf{CI\ R^{2}}$	$\mathbf{CI\ D}$	$\mathbf{R^{2}}$	$\mathbf{D}$	$\mathbf{R^{2}}$	$\mathbf{D}$	$\mathbf{CI\ R^{2}}$	$\mathbf{CI\ D}$
Airfoil Noise	0.56	0.15	0.53 – 0.58	0.14 – 0.16	0.00 – 0.18	0.12 – 0.47	0.43	0.50	0.33 – 0.51	0.49 – 0.50
Concrete Strength	0.75	0.12	0.73 – 0.77	0.11 – 0.13	0.00 – 0.38	0.09 – 0.44	0.73	0.50	0.66 – 0.79	0.49 – 0.50
Energy Efficiency	0.91	0.16	0.89 – 0.93	0.15 – 0.17	0.10 – 0.75	0.11 – 0.46	0.91	0.50	0.87 – 0.93	0.49 – 0.50
Yacht	0.87	0.15	0.84 – 0.91	0.14 – 0.18	-0.30 – 0.15	0.03 – 0.50	0.86	0.50	0.78 – 0.92	0.49 – 0.50
Bike (Day)	0.95	0.12	0.92 – 0.98	0.11 – 0.14	-0.30 – 0.70	0.05 – 0.39	0.98	0.50	0.97 – 0.99	0.49 – 0.50

TABLE VI : Performance Comparison of ANFIS, MO-ANFIS and X-ANFIS Across 9 Datasets, Evaluated Using (

R^{2}

) and Distinguishability (

D

). 95% Confidence Intervals (CI) are Reported for

R^{2}

and

D

Fig. 3 : Comparison of X-ANFIS and MO-ANFIS. Top row presents Kernel Density Estimation heatmaps of MF centers for features temperature, pressure, humidity, vacuum, named as F1–F4. Bottom row displays the corresponding MF shapes for F2, with red circles indicating center locations.

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Adaptive Neuro-Fuzzy Inference System (ANFIS) was designed to combine the learning capabilities of neural network with the reasoning transparency of fuzzy logic. However, conventional ANFIS architectures suffer from structural complexity, where the product-based inference mechanism causes an exponential explosion of rules in high-dimensional spaces. We herein propose the Kolmogorov-Arnold Neuro-Fuzzy Inference System (KANFIS), a compact neuro-symbolic architecture that unifies fuzzy reasoning with additive function decomposition. KANFIS employs an additive aggregation mechanism, under which both model parameters and rule complexity scale linearly with input dimensionality rather than exponentially. Furthermore, KANFIS is compatible with both Type-1 (T1) and Interval Type-2 (IT2) fuzzy logic systems, enabling explicit modeling of uncertainty and ambiguity in fuzzy representations. By using sparse masking mechanisms, KANFIS generates compact and structured rule sets, resulting in an intrinsically interpretable model with clear rule semantics and transparent inference processes. Empirical results demonstrate that KANFIS achieves competitive performance against representative neural and neuro-fuzzy baselines.

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