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A Kolmogorov-Arnold Neural Model for Cascading Extremes

Miguel de Carvalho, Clemente Ferrer, Ronny Vallejos

TL;DR

The paper tackles cascading extreme events by merging Extreme Value Theory with Kolmogorov–Arnold Networks to learn a covariate-dependent Probability of Cascade surface $\alpha_I(\mathbf{x})$. It introduces the KANE architecture, a three-layer KAN with a final activation to constrain outputs to $[0,1]$, and models inner/outer functions with splines to approximate the unknown Kolmogorov representation. It also develops extensions to multi-trigger, multicategory, ordinal, and continuous follow-ups, and provides model checking via Dunn–Smyth residuals, along with theoretical underpinnings based on a Kolmogorov superposition operator. Through artificial data experiments and two real-world applications (Earthquake–Tsunami and Tropical Cyclone–SST), the approach demonstrates accurate POC recovery, interpretability of spatial patterns, and robust diagnostic performance, offering a tool for extrapolating cascade risks in seismology and climatology.

Abstract

This paper addresses the growing concern of cascading extreme events, such as an extreme earthquake followed by a tsunami, by presenting a novel method for risk assessment focused on these domino effects. The proposed approach develops an extreme value theory framework within a Kolmogorov-Arnold network (KAN) to estimate the probability of one extreme event triggering another, conditionally on a feature vector. An extra layer is added to the KAN architecture to ensure that the parameter of interest lies within the unit interval, and we refer to the resulting neural model as KANE (KAN with Natural Enforcement). The proposed method is backed by exhaustive numerical studies and further illustrated with real-world applications to seismology and climatology.

A Kolmogorov-Arnold Neural Model for Cascading Extremes

TL;DR

The paper tackles cascading extreme events by merging Extreme Value Theory with Kolmogorov–Arnold Networks to learn a covariate-dependent Probability of Cascade surface . It introduces the KANE architecture, a three-layer KAN with a final activation to constrain outputs to , and models inner/outer functions with splines to approximate the unknown Kolmogorov representation. It also develops extensions to multi-trigger, multicategory, ordinal, and continuous follow-ups, and provides model checking via Dunn–Smyth residuals, along with theoretical underpinnings based on a Kolmogorov superposition operator. Through artificial data experiments and two real-world applications (Earthquake–Tsunami and Tropical Cyclone–SST), the approach demonstrates accurate POC recovery, interpretability of spatial patterns, and robust diagnostic performance, offering a tool for extrapolating cascade risks in seismology and climatology.

Abstract

This paper addresses the growing concern of cascading extreme events, such as an extreme earthquake followed by a tsunami, by presenting a novel method for risk assessment focused on these domino effects. The proposed approach develops an extreme value theory framework within a Kolmogorov-Arnold network (KAN) to estimate the probability of one extreme event triggering another, conditionally on a feature vector. An extra layer is added to the KAN architecture to ensure that the parameter of interest lies within the unit interval, and we refer to the resulting neural model as KANE (KAN with Natural Enforcement). The proposed method is backed by exhaustive numerical studies and further illustrated with real-world applications to seismology and climatology.
Paper Structure (15 sections, 4 theorems, 52 equations, 7 figures, 1 table)

This paper contains 15 sections, 4 theorems, 52 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Neural Modeling of Cascading Extremes
  3. The POC (Probability of Cascade) Surface
  4. KANE: KAN with Natural Enforcement
  5. Model Checking and Diagnostics
  6. Consequences and Extensions
  7. Multi-Trigger Systems
  8. Categorical, Ordinal, and Continuous Follow-up Events
  9. Numerical Experiments on Artificial Data
  10. Artificial Data Generating Processes and Preliminary Experiments
  11. Monte Carlo Evidence
  12. Empirical Examples
  13. Earthquake--Tsunami Data
  14. Tropical Cyclone--Sea Surface Temperature Data
  15. Implications and Future Directions

Key Result

Theorem 1

Let $f:[0, 1]^d \to \mathbb{R}$ be a continuous function. Then, $f$ can be expressed as follows for some continuous one-dimensional functions $\Phi_{i, j}^{(1)}$ and $\Phi_{j}^{(2)}$.

Figures (7)

  • Figure 1: Architecture of three-layer KANE model for the POC surface (binary follow-up event) illustrated on two-feature setting.
  • Figure 2: Single-sample experiments for Scenarios A, B, and C.
  • Figure 3: Monte Carlo means for Scenarios A, B, and C.
  • Figure 4: Point pattern of earthquakes (red) and associated tsunami occurrences (blue).
  • Figure 5: KANE POC surface estimate for Earthquake--Tsunami data, considering percentile 15, 40, 60, and 85 of depth.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 1: Kolmogorov's superposition theorem
  • Example 1: Tail dependence coefficient
  • Example 2: Extremal probabilistic index
  • Definition 1: POC Surface
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Definition 2: Multicategorical POC Surfaces
  • Theorem 3: Continuity of Kolmogorov superposition operator
  • ...and 2 more