Improved next generation reservoir computing with time decay factor and kernel function

Modeling and prediction of dynamical systems are essential in both scientific and engineering fields, but the nonlinearity and chaotic behavior present significant challenges to traditional methods. Next-generation reservoir computing (NGRC) aims to mitigate the randomness of conventional reservoir computing through time-delay feature construction, but it still faces issues such as relying on fixed nonlinear basis functions for feature mapping making it difficult to adapt to varying system dynamics, insufficient sensitivity to historical data, and high-dimensional computational complexity. In this paper, we propose an enhanced NGRC method that improves adaptability and predictive accuracy for complex dynamical systems by integrating temporal decay and kernel functions from the attention mechanism. The decay factor dynamically adjusts the weights of historical data through exponential decay, emphasizing recent temporal dependencies. Meanwhile, a Gaussian kernel function enhances nonlinear mapping capabilities, enabling the model to capture intricate dynamical patterns. Experiments on chaotic systems, including the Lorenz and double-scroll systems, demonstrate that the improved NGRC model significantly enhances prediction accuracy and stability, particularly in long-term forecasting scenarios. Moreover, it exhibits strong generalization capability with respect to variations in initial conditions. The proposed method offers valuable insights into the modeling and prediction of complex dynamical systems and shows great potential for future applications.