The internal model principle in reservoir computing
Abstract
The paper studies learning in reservoir computing systems through the lens of the internal model principle. The key idea is to model training signals as outputs of an exogenous signal generator, thus connecting reservoir computing to output regulation, observer theory, and frequency-domain interpolation. For linear signal generators, we characterize the steady-state response of a linear reservoir through invariant subspaces, establishing connections with Luenberger observers and classical frequency-domain interpolation. For nonlinear signal generators, we obtain analogous results using invariant manifolds, connecting synchronization to Kazantzis-Kravaris/Luenberger observers and readout training to nonlinear moment matching. Case studies illustrate how our perspective explains possible instability of trained predictors and informs reservoir design.
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BibTeX
@misc{saqib2026internal,
title = {{The internal model principle in reservoir computing}},
author = {Danyal Saqib and Alberto Padoan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08987},
eprint = {2610.08987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08987},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The internal model principle in reservoir computing
AU - Danyal Saqib
AU - Alberto Padoan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08987
UR - https://arxiv.org/abs/2610.08987
AB - The paper studies learning in reservoir computing systems through the lens of the internal model principle. The key idea is to model training signals as outputs of an exogenous signal generator, thus connecting reservoir computing to output regulation, observer theory, and frequency-domain interpolation. For linear signal generators, we characterize the steady-state response of a linear reservoir through invariant subspaces, establishing connections with Luenberger observers and classical frequency-domain interpolation. For nonlinear signal generators, we obtain analogous results using invariant manifolds, connecting synchronization to Kazantzis-Kravaris/Luenberger observers and readout training to nonlinear moment matching. Case studies illustrate how our perspective explains possible instability of trained predictors and informs reservoir design.
ER -