KANFIS: A Neuro-Symbolic Framework for Interpretable and Uncertainty-Aware Learning

Binbin Yong; Haoran Pei; Jun Shen; Haoran Li; Qingguo Zhou; Zhao Su

cs.AIFeb 3, 2026

KANFIS: A Neuro-Symbolic Framework for Interpretable and Uncertainty-Aware Learning

Authors: Binbin Yong, Haoran Pei, Jun Shen, Haoran Li, Qingguo Zhou, Zhao Su

Organizations: School of Information Science and Engineering, Lanzhou University, China · School of Computing and Information Technology, University of Wollongong, Australia · Department of Data Science and Artificial Intelligence, Monash University, Australia

Abstract

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.

Figures & tables

Figure 1 : Limitations of Existing Interpretable Models. Conventional ANFIS and KAN models suffer from several limitations in terms of interpretability and model complexity.

Figure 2 : Top: KANFIS structures with one and two layers, respectively. Dashed lines indicate paths that may be selected during learning. Bottom: Computational flow of the model. $x$ denotes the input features, $y$ the predicted output, and $\omega$ the weight of each rule, reflecting the influence of each rule pattern on the final prediction.

Dataset	Metrics	IT2-KANFIS	T1-KANFIS	MLP	T1-ANFIS	IT2-ANFIS	KAN
CCPP	MAPE	0.7047	0.6777	0.7164	0.6759	0.6696	0.6389
	RMSE	4.1240	3.9542	4.1883	3.9980	4.0047	3.9313
	MAE	3.1975	3.0760	3.2553	3.0688	3.0439	2.8988
Parkinsons	MAPE	14.891	14.477	19.604	16.167	15.446	14.610
	RMSE	0.0397	0.0405	0.0527	0.0449	0.0433	0.0404
	MAE	0.0291	0.0289	0.0389	0.0320	0.0310	0.0293

Table 1 : Comparative experimental results. We adopt a clustering-based IT2-ANFIS to ensure its suitability as a baseline under high-dimensional feature settings. The symbol ‘-’ indicates that the model cannot be trained due to the curse of dimensionality.

IF	THEN	Interpretation
AT is HIGH	$-0.3407\cdot(7.26\cdot\mathcal{M}_{\text{AT}}(x_{\text{AT}}))$	High ambient temperature reduces air density and mass flow, lowering power output ( Saravanamuttoo et al., 2001 ) .
V is LOW & AP is MED & RH is HIGH	$0.3070\cdot(2.96\cdot\mathcal{M}_{\text{V}}(x_{\text{V}})+0.14\cdot\mathcal{M}_{\text{AP}}(x_{\text{AP}})+0.16\cdot\mathcal{M}_{\text{RH}}(x_{\text{RH}}))$	Low exhaust vacuum and high relative humidity reduce back pressure and increase specific heat, creating optimal operating conditions ( Cengel and Boles, 2002 ) .
AT is MED & AP is HIGH	$0.2541\cdot(3.38\cdot\mathcal{M}_{\text{AT}}(x_{\text{AT}})+0.48\cdot\mathcal{M}_{\text{AP}}(x_{\text{AP}}))$	Higher ambient pressure increases air density, enhancing turbine intake flow and efficiency even at moderate temperatures ( Saravanamuttoo et al., 2001 ) .
V is LOW & RH is LOW	$0.2038\cdot(2.31\cdot\mathcal{M}_{\text{V}}(x_{\text{V}})+1.61\cdot\mathcal{M}_{\text{RH}}(x_{\text{RH}}))$	Low exhaust vacuum dominates efficiency; reduced condenser back pressure ensures high power output even at low humidity ( Moran et al., 2010 ) .
AT is MED & V is HIGH	$-0.1831\cdot(3\cdot\mathcal{M}_{\text{AT}}(x_{\text{AT}})+0.53\cdot\mathcal{M}_{\text{V}}(x_{\text{V}}))$	High exhaust vacuum increases condenser back pressure, restricting steam expansion and offsetting medium-temperature benefits ( Moran et al., 2010 ) .
AT is HIGH & RH is LOW	$-0.1351\cdot(3.88\cdot\mathcal{M}_{\text{AT}}(x_{\text{AT}})+0.93\cdot\mathcal{M}_{\text{RH}}(x_{\text{RH}}))$	Low relative humidity implies lower specific heat, slightly reducing gas turbine output at similar temperatures ( Cengel and Boles, 2002 ) .

Table 2 : Interpretability analysis of fuzzy rules on the CCPP dataset. The six most important rules learned by the model and their associated physical principles are presented in the table, with their validity supported by evidence from the cited relevant physical studies.

Figure 3 : Comparison of the average number of features per rule across datasets. The figure illustrates the comparison between the number of features used in the rules and the original number of features in each dataset, depending on whether regularization is applied.

IF	THEN	Interpretation
SystolicBP is HIGH & DiastolicBP is MED	$P_{highrisk}=4.0611\cdot(7.28\cdot\mathcal{M}_{\text{SystolicBP}}(x_{\text{SystolicBP}})+2.09\cdot\mathcal{M}_{\text{DiastolicBP}}(x_{\text{DiastolicBP}}))$	Elevated systolic and diastolic BP increase cardiovascular risk ( He and Whelton, 1999 ) .
Age is LOW & SystolicBP is HIGH & BS is LOW	$P_{lowrisk}=0.814\cdot(3.59\cdot\mathcal{M}_{\text{Age}}(x_{\text{Age}})+0.94\cdot\mathcal{M}_{\text{SystolicBP}}(x_{\text{SystolicBP}})+1.56\cdot\mathcal{M}_{\text{BS}}(x_{\text{BS}}))$	Young age with slightly high BP and low glucose indicates low overall risk ( D’Agostino et al., 2008 ) .
DiastolicBP is LOW	$P_{highrisk}=0.2811\cdot(4.49\cdot\mathcal{M}_{\text{DiastolicBP}}(x_{\text{DiastolicBP}}))$	Low diastolic BP may reduce perfusion and raise cardiovascular risk ( McEvoy et al., 2016 ) .
SystolicBP is HIGH & BS is LOW	$P_{lowrisk}=1.4001\cdot(1.48\cdot\mathcal{M}_{\text{SystolicBP}}(x_{\text{SystolicBP}})+3.81\cdot\mathcal{M}_{\text{BS}}(x_{\text{BS}}))$ $P_{mediumrisk}=1.1433\cdot(1.48\cdot\mathcal{M}_{\text{SystolicBP}}(x_{\text{SystolicBP}})+3.81\cdot\mathcal{M}_{\text{BS}}(x_{\text{BS}}))$	Slightly high SBP with low glucose corresponds to low-to-moderate risk ( D’Agostino et al., 2008 ) .
SystolicBP is MED & BS is LOW & BodyTemp is LOW	$P_{mediumrisk}=1.0646\cdot(2.2\cdot\mathcal{M}_{\text{SystolicBP}}(x_{\text{SystolicBP}})+3.21\cdot\mathcal{M}_{\text{BS}}(x_{\text{BS}})+1.34\cdot\mathcal{M}_{\text{BodyTemp}}(x_{\text{BodyTemp}}))$	High SBP with low glucose and normal/low temp indicates moderate risk ( D’Agostino et al., 2008 ) .
Age is HIGH	$P_{highrisk}=-0.3013\cdot(2.81\cdot\mathcal{M}_{\text{Age}}(x_{\text{Age}}))$	Advanced age independently increases cardiovascular risk ( D’Agostino et al., 2008 ) .

Table 3 : Interpretability analysis of fuzzy rules on the MHR dataset. The table presents the six most important rules learned by the model along with their corresponding medical principles, and their validity is supported by evidence from the cited relevant studies. For clarity, only the class formula that best matches each rule is reported in the THEN clause.

Dataset	Metric	Normal Regularization	High Regularization	Low Regularization
CCPP	RMSE	4.1240	4.2304	4.1903
Parkinsons	RMSE	0.0397	0.0418	0.0403
BCW	Acc	0.9912	0.9474	0.9474

Table 4: Impact of Different Levels of Regularization on Model Accuracy.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Dimension	T1-KANFIS (Acc)	IT2-KANFIS
APS Failure at Scania Trucks	171	0.9898	0.9900
Dexter	20000	0.8667	0.9333
ForestCover	54	0.9420	0.9491

Appendix

Table 5: Accuracy of KANFIS on the supplementary datasets. The three additional datasets reported in the table are all highly high-dimensional datasets. Among them, the ForestCover dataset was specifically included due to its extremely large scale, containing 581,012 instances.

Dataset	Metric	Ours	NAMs	GAMI-NET	NODE-GAM
Parkinsons	RMSE	0.0397	0.0408	0.0403	–
BCW	Acc	0.9912	0.9651	0.9649	0.9737

Appendix

Table 6: Performance comparison with other interpretable models. The table presents comparative results with three representative interpretable models, namely NAMs, GAMI-NET, and NODE-GAM. Due to limitations in the available open-source implementation, NODE-GAM was unable to produce results on the Parkinsons dataset.

Dataset	IT2-KANFIS	T1-KANFIS	MLP	ANFIS	KAN
CCPP	0.3914, 4.1240	0.3560, 3.9542	0.0902, 4.1883	0.3042, 3.9980	4.8063, 3.9313
BCW	0.0212, 0.9912	0.0207, 0.9912	0.0067, 0.9912	–	1.4943, 0.9736
Spam	0.1357, 0.9392	0.1175, 0.9381	0.0475, 0.9175	–	–

Appendix

Table 7: Time-Accuracy Comparison of Models. The training time is measured in seconds per training epoch. For regression datasets, accuracy is measured using RMSE, while for classification datasets, accuracy is measured using Acc. Values are reported as RMSE and Accuracy.

Dataset	IT2-KANFIS	T1-KANFIS	MLP	ANFIS	KAN
CCPP	521, 2600	361, 1640	193, 480	429, 1274	1600, 9972
BCW	4822, 24330	3986, 18396	1058, 2208	5139, 15398	2240, 11518
Spam	7314, 36960	5034, 23280	1922, 3936	–	–

Appendix

Table 8: Comparison of Parameter Count and FLOPs for Different Models. Values are reported as the number of parameters and FLOPs.

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Aug 12, 2026, cs.AIJ/K move · Enter open · S save

HyperANFIS: Enhancing Rule Representation and Interpretability in Adaptive Neuro-Fuzzy Systems via Hyperbolic Geometry

Haoran Pei, Zhao Su, Zetao Lin, Haoran Li+5

school of Information Science and Engineering, Lanzhou University · Department of Data Science and AI, Monash University · School of Computing and Information Technology, University of Wollongong+1

Fuzzy Hyperbolic Learning Hyperbolic Space Interpretability

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The adaptive neuro-fuzzy inference system (ANFIS) is an interpretable reasoning framework capable of generating explicit IF-THEN fuzzy rules, making it suitable for tasks requiring transparent reasoning. However, existing ANFIS models generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. To address this issue, we propose Hyperbolic ANFIS (HyperANFIS), a hyperbolic extension of ANFIS. HyperANFIS preserves the fuzzy semantics and core architecture of conventional ANFIS while performing rule-prototype learning, rule activation, and consequent aggregation in hyperbolic space. It also retains the ability to generate interpretable IF-THEN rules. By exploiting the representational properties of hyperbolic geometry, HyperANFIS strengthens the fuzzy inference process, thereby improving predictive accuracy, inter-rule collaboration, and the credibility of its interpretable rules. Experimental results show that HyperANFIS consistently outperforms the standard ANFIS baseline and various ANFIS variants across all datasets, while also generating higher-quality fuzzy rules.

Abstract

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HyperANFIS: Enhancing Rule Representation and Interpretability in Adaptive Neuro-Fuzzy Systems via Hyperbolic Geometry

SparseKAN: Compressing Kolmogorov--Arnold Networks Across Basis Functions, Neurons, and Bits

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