Uncertainty Quantification of Next Generation Reservoir Computing with Applications to Memory-Driven Dynamical Systems

2026-09-26

AI Summary

This study addresses the absence of uncertainty quantification methods for Next-Generation Reservoir Computing (NGRC) in memory-driven systems by integrating Bayesian ridge regression with conformal prediction theory to systematically investigate NGRC uncertainty estimation mechanisms. By elucidating how residual distributions and regularization influence prediction intervals, we establish a theoretical framework bridging low- and high-dimensional settings, which is further extended to quadratic feature mappings. Numerical simulations validate the theoretical predictions, clarifying the mechanistic roles of dimensionality, regularization, and distribution shift in calibration efficiency. Ultimately, this work provides reliable criteria for method selection in dynamical systems, offering a principled approach to quantifying predictive uncertainty within the NGRC paradigm.

Abstract

Nonlinear dynamical systems with memory arise across science and engineering, yet uncertainty quantification for efficient forecasting methods such as Next Generation Reservoir Computing (NGRC) remains underdeveloped. We study Bayesian ridge and conformal prediction intervals for NGRC and characterize when their uncertainty estimates agree or differ. In low dimensions, their asymptotic widths are governed by different summaries of the residual distribution, so agreement depends on residual shape rather than dimensionality alone. In high dimensions, regularization introduces a further tradeoff between estimation variance, shrinkage bias, and posterior uncertainty, leading to an explicit transition between regimes where Bayesian intervals are wider or narrower than conformal intervals. We extend these results to quadratic NGRC feature maps and give sufficient conditions for transferring the analysis to temporally dependent forecast windows. Simulations and real-data experiments support the theoretical predictions and illustrate how residual distribution, regularization, dimensionality, and distribution shift affect interval calibration and efficiency. These results provide a principled framework for choosing and interpreting uncertainty quantification methods in reservoir-based forecasting.

Problem

• Uncertainty Quantification
• Next Generation Reservoir Computing
• Nonlinear Dynamical Systems
• Conformal Prediction
• Bayesian Ridge Regression

Innovation

• Next Generation Reservoir Computing
• Uncertainty Quantification
• Conformal Prediction
• Bayesian Ridge Regression
• Memory-Driven Dynamical Systems