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

TL;DR

This paper introduces the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel neuro-symbolic architecture that models non-monotonic causal relationships using learnable B-spline functions, enhancing interpretability and accuracy.

Contribution

The paper proposes KA-FCM, a new architecture that replaces scalar weights with B-spline functions based on the Kolmogorov-Arnold theorem, enabling non-monotonic causal modeling without added complexity.

Findings

KA-FCM outperforms standard FCM in non-monotonic inference, symbolic regression, and chaotic time-series forecasting.

KA-FCM achieves accuracy comparable to MLPs while maintaining interpretability.

Experimental results validate KA-FCM's ability to explicitly extract mathematical laws from learned causal edges.

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.