Kolmogorov-Arnold Reservoir Computing

2026-06-18

AI Summary

Traditional reservoir computing struggles to capture long-range dependencies in dynamical system modeling and often faces a trade-off between limited expressivity and explosive feature dimensionality. This work proposes Kolmogorov–Arnold Reservoir Computing (KARC), which, for the first time, integrates the Kolmogorov–Arnold representation theorem into the reservoir framework. By replacing stochastic recurrent structures with explicit basis function expansions, KARC enables recursion-free, closed-form linear training. The method achieves high representational capacity while substantially reducing computational overhead and seamlessly integrates with diffusion models. Experiments demonstrate that KARC outperforms existing reservoir approaches on challenging dynamical system prediction tasks, including partial differential equations, and is successfully applied to text-to-image generation.

Abstract

Reservoir computing offers a lightweight framework for forecasting dynamical systems but may struggle to capture long-range dependencies due to limited representational capacity. Conventional reservoir computing recurrently uses trainable reservoirs with hyperparameter sensitivity, while the next-generation reservoir computing removes recurrence at the cost of rapidly growing feature dimensions. Here, we develop Kolmogorov-Arnold Reservoir Computing (KARC), which replaces reservoirs with explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem. We rigorously show that KARC is a lightweight design of Kolmogorov-Arnold networks (KANs), preserving the potential expressive capacity of KANs while admitting efficient closed-form training of reservoir computing. At comparable cost, KARC outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations. It can also be integrated with generative diffusion models for text-to-image generation. This work thus establishes a principled bridge between reservoir computing and KANs, enabling efficient and high-fidelity dynamical system forecasting.

Problem

• reservoir computing
• long-range dependencies
• dynamical systems
• Kolmogorov-Arnold representation
• representational capacity

Innovation

• Kolmogorov-Arnold representation
• reservoir computing
• basis-function expansion
• closed-form training
• dynamical system forecasting