X-ANFIS: Explainability in Neuro-Fuzzy Systems

• X-ANFIS is a neuro-fuzzy framework that integrates explicit explainability constraints to yield linguistically interpretable rules alongside high predictive performance.
• It employs a differentiable objective based on membership function distinguishability, ensuring well-separated, non-overlapping fuzzy partitions.
• An alternating bi-objective gradient optimization recovers non-convex trade-offs between accuracy and interpretability across diverse datasets.

Explainability constraints in adaptive neuro-fuzzy inference systems (X-ANFIS) refer to the explicit incorporation of interpretability as an integral objective in the end-to-end learning process, aligning system parameters to facilitate semantically meaningful, human-comprehensible fuzzy rules without sacrificing predictive performance. The X-ANFIS methodology defines a concrete, differentiable explainability objective—distinguishability of membership functions—and utilizes alternating bi-objective gradient optimization. This framework enables the recovery of accurate and linguistically interpretable rule bases in otherwise non-convex segments of the accuracy–explainability Pareto front (Khaled et al., 22 Feb 2026).

1. Accuracy–Explainability Trade-Off in Neuro-Fuzzy Systems

The foundational challenge in neuro-fuzzy models arises from the tension between transparency (linguistic clarity of rules) and statistical accuracy. Fuzzy systems achieve intuitive explainability via rule-based architectures and linguistic variables, imparting their outputs with human-understandable semantics. However, enforcing interpretability through strong constraints (few rules, well-separated membership functions) often comes at a direct cost to accuracy, as models may lose expressiveness or coverage. Conversely, unconstrained, high-capacity ANFIS models optimized solely for loss can fine-tune thousands of parameters—yielding highly overlapping, semantically ambiguous fuzzy sets where linguistic rule interpretation is compromised.

This dilemma, frequently described as the "accuracy–explainability trade-off," motivates the development of explicit explainability constraints as part of the training process, rather than relying on post-hoc pruning or computationally intensive evolutionary multi-objective optimization (Khaled et al., 22 Feb 2026).

2. X-ANFIS Model Structure and Semantic Initialization

X-ANFIS operates on a standard zero-order Takagi–Sugeno ANFIS framework with $R$ fuzzy rules and $m$-dimensional real-valued input vectors $x=(x_1,\dots,x_m)$. The architecture comprises five layers:

1. Input layer: Passes $x_i$ directly.
2. Membership layer: Evaluates fuzzy membership degrees using Cauchy membership functions,

$$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$

where $c_{ij}$ and $\gamma_{ij}$ are the center and scale for the $j$th fuzzy set on the $i$th input.

1. Rule layer: Computes rule firing strengths,

$$w_j(x; \theta) = \prod_{i=1}^m \mu_{ij}(x_i; c_{ij}, \gamma_{ij})$$

2. Normalization layer: Normalizes firing strengths,

$m$0

3. Output layer: Weighted sum to obtain the final prediction,

$m$1

where $m$2 are zero-order consequent parameters.

Semantic initialization utilizes fuzzy c-means (FCM) clustering independently on each feature to obtain $m$3 cluster centroids and spreads, yielding antecedent parameters $m$4 that uniformly partition the input space in $m$5. This enhances rule interpretability and provides a stable starting point for gradient-based learning (Khaled et al., 22 Feb 2026).

3. Differentiable Explainability Objective: Distinguishability

A core innovation of X-ANFIS is the operationalization of distinguishability as a differentiable constraint during training. For each feature $m$6 and each adjacent pair of fuzzy sets $m$7 (those with consecutive ordered FCM centers), define the pairwise distance:

$m$8

A target distinguishability $m$9 (typically $x=(x_1,\dots,x_m)$0) is specified to maximize semantic separation without loss of coverage.

The explainability loss is a quadratic penalty, summed over all features and their adjacent fuzzy sets:

$x=(x_1,\dots,x_m)$1

where $x=(x_1,\dots,x_m)$2 denotes adjacent index pairs for feature $x=(x_1,\dots,x_m)$3. Gradients of $x=(x_1,\dots,x_m)$4 with respect to centers $x=(x_1,\dots,x_m)$5 are used to adjust membership function placement, holding $x=(x_1,\dots,x_m)$6 fixed in the X-pass to prevent degenerate solutions (e.g., reducing widths to artificially inflate $x=(x_1,\dots,x_m)$7) (Khaled et al., 22 Feb 2026).

4. Alternating Bi-Objective Gradient Optimization

To address the inability of scalarized objectives (e.g., $x=(x_1,\dots,x_m)$8) to recover non-convex Pareto-optimal trade-offs, X-ANFIS employs an alternating bi-objective gradient scheme:

• Performance Pass (Backward Pass):
  • Minimize mean squared error (MSE) of outputs.
  • Consequent parameters $x=(x_1,\dots,x_m)$9 are solved in closed form (regularized least squares).
  • Membership parameters $x_i$0 are updated via gradients. Cauchy function derivatives are

  $x_i$1

• Explainability Pass (X-pass):

  • Temporarily hold $x_i$2 and consequents fixed.
  • Update centers $x_i$3 by gradient descent on $x_i$4.

Pseudocode per epoch:

1. Forward pass to compute all outputs.
2. Backward (performance) pass: update $x_i$5 and solve $x_i$6.
3. X-pass: update only $x_i$7 minimizing $x_i$8.
4. Early stopping based on validation MSE.

This alternation enables X-ANFIS to converge to solutions with both competitive accuracy and interpretable, well-separated fuzzy sets, including points outside the convex hull attainable by scalarization or traditional MOO (Khaled et al., 22 Feb 2026).

5. Empirical Validation and Comparative Analysis

Extensive experimentation was performed on nine UCI regression datasets in approximately 5,000 configurations. Key outcomes:

• Initialization stability: Cauchy ANFIS maintains both accuracy ($x_i$9) and distinguishability ($$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$0) across a wide range of $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$1, while Gaussian ANFIS collapses for small initial spreads.
• Pareto front recovery: Weighted-scalarization MO-ANFIS yields solutions on the convex hull of the accuracy–explainability graph; X-ANFIS recovers non-convex trade-off points with $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$2 and $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$3 within $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$4–$$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$5 of pure ANFIS in 8 out of 9 datasets using $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$6 rules.
• Partition visualization: X-ANFIS achieves uniform, evenly-spaced partitions, in contrast to MO-ANFIS, which clusters partitions around dense areas and leaves gaps (violating coverage).
• Semantic plane analysis: Only X-ANFIS consistently yields membership function plots that satisfy consistency, completeness, and desired distinguishability.

A summary table of design components is provided below.

Component	Method/Constraint	Role in Explainability
Membership Function	Cauchy; semantically spread via FCM	Stability, well-partitioned, interpretable
Explainability Loss	$$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$7 on $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$8 separation	Enforces distinguishable, non-overlapping sets
Optimization Scheme	Alternating performance and X-passes	Recovers non-convex trade-offs

6. Computational Considerations, Limitations, and Extensions

Compared to evolutionary MOO approaches, X-ANFIS is $$\mu_{ij}(x_i; c_{ij}, \gamma_{ij}) = \frac{1}{1 + \left(\frac{x_i - c_{ij}}{\gamma_{ij}}\right)^2}$$9–$c_{ij}$0 faster as it avoids population-based search, relying on efficient gradient updates. Scalarization methods lack coverage over non-convex regions, while the alternating approach improves search diversity. As $c_{ij}$1 increases ($c_{ij}$2), the fixed universe $c_{ij}$3 imposes a tension: to maintain $c_{ij}$4, coverage begins to conflict with distinguishability. This suggests that joint optimization over rule count ($c_{ij}$5) and distinguishability ($c_{ij}$6) would enhance model efficacy. Future extensions include adapting to Mamdani-type inference, incorporating structure learning for adaptive rule cardinality, and formal convergence analysis of the alternating scheme (Khaled et al., 22 Feb 2026).

7. Relationship to Neural Explainability-Constraint Paradigms

X-ANFIS is part of a broader class of explainability-constraint frameworks, including xNN—explainable neural networks—that enforce interpretability through explicit architecture constraints such as sparsity, orthogonality, and smoothness (Yang et al., 2019). While xNN achieves interpretability via additive sparse subnetworks, projection pursuit with orthogonality, and roughness penalization, X-ANFIS instantiates explainability within fuzzy rule induction and partition integrity. Both approaches emphasize constraints imposed during training—rather than post-hoc analysis or black-box explanation—to produce inherently transparent, trustworthy systems.

In summary, explainability constraints in X-ANFIS operationalize semantic interpretability as a mathematically well-posed training objective. Alternating between accuracy and distinguishability optimizations, combined with stable Cauchy membership functions and semantic initialization, yields adaptive neuro-fuzzy systems that navigate the accuracy–explainability trade-off and populate otherwise inaccessible regions of the Pareto frontier.