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

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

TL;DR

KANFIS addresses the interpretability and uncertainty limitations of traditional neuro-fuzzy and deep models by embedding fuzzy reasoning within a Kolmogorov-Arnold additive framework. It replaces fixed linear weights with learnable univariate membership functions on edges, supports both Type-1 and Interval Type-2 fuzzy logic, and stacks these fuzzy-functional layers to form a deep, yet compact, interpretable network. Regularization promotes sparsity and distinctiveness to yield concise, human-readable rules while preserving predictive accuracy, and its IT2 formulation enhances uncertainty handling. Empirical results show competitive performance against neural and neuro-fuzzy baselines, with explicit interpretability demonstrated through domain-aligned rules on CCPP and MHR, and ablation confirms the benefits of sparsity and rule diversity for readability.

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.

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

TL;DR

KANFIS addresses the interpretability and uncertainty limitations of traditional neuro-fuzzy and deep models by embedding fuzzy reasoning within a Kolmogorov-Arnold additive framework. It replaces fixed linear weights with learnable univariate membership functions on edges, supports both Type-1 and Interval Type-2 fuzzy logic, and stacks these fuzzy-functional layers to form a deep, yet compact, interpretable network. Regularization promotes sparsity and distinctiveness to yield concise, human-readable rules while preserving predictive accuracy, and its IT2 formulation enhances uncertainty handling. Empirical results show competitive performance against neural and neuro-fuzzy baselines, with explicit interpretability demonstrated through domain-aligned rules on CCPP and MHR, and ablation confirms the benefits of sparsity and rule diversity for readability.

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.
Paper Structure (31 sections, 4 theorems, 30 equations, 3 figures, 3 tables)

This paper contains 31 sections, 4 theorems, 30 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Intrinsic Interpretability and Additive Models
  4. Adaptive Neuro-Fuzzy Inference System (ANFIS)
  5. Kolmogorov-Arnold Network (KAN)
  6. Theoretical Connection: Splines and Fuzzy Sets
  7. Methodology
  8. Learnable Membership Functions
  9. Functional Edges as Fuzzy Rule Antecedents
  10. Kolmogorov-Arnold Aggregation Over Edges
  11. Deep Stacking of Fuzzy-Functional Layers
  12. Defuzzification via Linear Projection
  13. Regularization
  14. Entropy-based Antecedent Sparsity.
  15. Rule Distinctiveness.
  16. ...and 16 more sections

Key Result

Lemma 1.1

Let $\psi(x)$ be any continuous function defined on a compact set $C \subset \mathbb{R}$. For any $\epsilon_1 > 0$, the KANFIS edge function $\phi_{ij}(x)$ can be parameterized by adjusting its centers and widths $\{\mu, \sigma_l, \sigma_u\}$ such that it approximates $\psi(x)$ within the error tole

Figures (3)

  • 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.
  • 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.

Theorems & Definitions (12)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • proof
  • Definition 2.1: Product Fuzzy System (PFS) Structure
  • Definition 2.2: Additive Fuzzy System (AFS) Structure
  • Theorem 2.3: Universal Approximation Equivalence
  • proof
  • Theorem 2.4: Efficiency Disparity in High Dimensions
  • proof
  • ...and 2 more