# A mathematical framework for time-delay reservoir computing analysis

arXiv: https://arxiv.org/abs/2603.18706 · HTML: https://mlsanity.com/paper/2603.18706/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 19 Mar 2026.

## Builds on (most cited first, 5 of 5)

- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Deep KKL: Data-driven Output Prediction for Non-Linear Systems](https://mlsanity.com/paper/2103.12443/) (2021)
- [The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions](https://mlsanity.com/paper/2107.11363/) (2021)
- [The ISS framework for time-delay systems: a survey](https://mlsanity.com/paper/2206.06167/) (2022)
- [Separation capacity of linear reservoirs with random connectivity matrix](https://mlsanity.com/paper/2404.17429/) (2024)

## Abstract (arXiv)

Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.