Non-monotonic causal discovery with Kolmogorov-Arnold Fuzzy Cognitive Maps

Jose L. Salmeron

cs.AIApr 6, 2026

Non-monotonic causal discovery with Kolmogorov-Arnold Fuzzy Cognitive Maps

Authors: Jose L. Salmeron

Organizations: School of Engineering, CUNEF Universidad, Madrid, Spain

Abstract

Fuzzy Cognitive Maps constitute a neuro-symbolic paradigm for modeling complex dynamic systems, widely adopted for their inherent interpretability and recurrent inference capabilities. However, the standard FCM formulation, characterized by scalar synaptic weights and monotonic activation functions, is fundamentally constrained in modeling non-monotonic causal dependencies, thereby limiting its efficacy in systems governed by saturation effects or periodic dynamics. To overcome this topological restriction, this research proposes the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel architecture that redefines the causal transmission mechanism. Drawing upon the Kolmogorov-Arnold representation theorem, static scalar weights are replaced with learnable, univariate B-spline functions located on the model edges. This fundamental modification shifts the non-linearity from the nodes' aggregation phase directly to the causal influence phase. This modification allows for the modeling of arbitrary, non-monotonic causal relationships without increasing the graph density or introducing hidden layers. The proposed architecture is validated against both baselines (standard FCM trained with Particle Swarm Optimization) and universal black-box approximators (Multi-Layer Perceptron) across three distinct domains: non-monotonic inference (Yerkes-Dodson law), symbolic regression, and chaotic time-series forecasting. Experimental results demonstrate that KA-FCMs significantly outperform conventional architectures and achieve competitive accuracy relative to MLPs, while preserving graph- based interpretability and enabling the explicit extraction of mathematical laws from the learned edges.

Figures & tables

Panel of Figure 1 — Fig. 1: Causal relationship comparison

Fig. 2: Kolmogorov-Arnold Fuzzy Cognitive Map architecture.

Fig. 3: Comparative modeling of the non-monotonic Yerkes-Dodson law. The scatter plot represents the synthetic ground truth exhibiting an inverted U-shape. The dashed blue line indicates the standard FCM prediction. The green dashed line shows the MLP results. The solid red line shows the function learned by the KA-FCM edge.

Model	MSE	Max. Abs. Error	Std. Dev. Error
Standard FCM	$3.96\times 10^{-1}$	$0.971$	$0.556$
MLP	$1.18\times 10^{-4}$	$0.025$	$0.009$
KA-FCM	$\mathbf{4.10\times 10^{-5}}$	$\mathbf{0.015}$	$\mathbf{0.006}$

TABLE I: Performance and stability analysis (Experiment I)

Parameter	Optimal value	Correlation w/ Error
Grid size ( $G$ )	$4$	$+0.05$ (minimal)
Learning rate ( $\eta$ )	$0.1$	$\mathbf{-0.61}$ (critical)
Epochs	$611$	$-0.16$ (low)

TABLE II: Hyperparameter sensitivity (Experiment I)

Model	MSE	Max. Abs. Error	Std. Dev. Error
Standard FCM	$4.26\times 10^{-1}$	$1.119$	$0.654$
MLP (Baseline)	$4.37\times 10^{-4}$	$0.082$	$0.011$
KA-FCM	$\mathbf{3.40\times 10^{-8}}$	$\mathbf{0.0004}$	$\mathbf{0.0002}$

TABLE III: Performance and stability analysis (Experiment II)

Fig. 4: Learned edge functions comparison.

Parameter	Optimal value	Correlation w/ Error
Grid size ( $G$ )	$19$	$\mathbf{-0.32}$ (significant)
Learning rate ( $\eta$ )	$0.1$	$\mathbf{-0.40}$ (high)
Epochs	$1500$	$-0.22$ (moderate)

TABLE IV: Hyperparameter sensitivity (Experiment II)

Model	MAPE (%)	Max. Abs. Error	Std. Dev. Error
Standard FCM	$41.37\%$	$0.162$	$0.058$
MLP	$27.67\%$	$0.097$	$0.027$
KA-FCM	$\mathbf{13.66\%}$	$\mathbf{0.072}$	$\mathbf{0.023}$

TABLE V: Performance and stability analysis (Experiment III - Mackey-Glass)

Fig. 5: One-step-ahead forecasting performance on the chaotic Mackey-Glass time series ( $\tau=17$ ).

Parameter	Optimal value	Correlation w/ Error (MAPE)
Grid size ( $G$ )	$19$	$-0.07$ (slight)
Learning rate ( $\eta$ )	$0.05$	$\mathbf{-0.62}$ (critical)
Epochs	$1277$	$-0.28$ (moderate)

TABLE VI: Hyperparameter sensitivity (Experiment III)

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Causal discovery from observational tabular data remains fundamentally challenging, primarily due to the heterogeneity of underlying causal mechanisms and the high-dimensional combinatorial search space of Directed Acyclic Graphs (DAGs). In this paper, we propose DAG-FM, a novel foundation model architecture that amortizes causal discovery. Unlike direct matrix prediction, DAG-FM decomposes the causal discovery process into two auto-regressive stages using two specialized Transformer-based sub-modules: a leaf-node predictor and a parent-node predictor. To effectively model complex row-column interactions, we adopt a robust tabular interaction block to output feature-wise representations. Crucially, to handle diverse and unknown Functional Causal Model (FCM) assumptions in real-world scenarios, we introduce Mixture-of-Leaf-Experts (MoLE), allowing the model to dynamically route and adapt to identifiable mechanism families. Through an iterative inference algorithm, DAG-FM seamlessly extracts causal orderings and constructs valid DAGs. Extensive experiments demonstrate that DAG-FM achieves state-of-the-art performance on both synthetic benchmarks and complex real-world datasets, significantly outperforming traditional classical algorithms and recent foundation models in both accuracy and scalability.

Abstract

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