# Simple Cycle Reservoirs are Universal

arXiv: https://arxiv.org/abs/2308.10793 · HTML: https://mlsanity.com/paper/2308.10793/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2023.

## Built on by (most cited first, 8 of 8)

- [Stochastic Reservoir Computers](https://mlsanity.com/paper/2405.12382/) (2024)
- [Universality of Real Minimal Complexity Reservoir](https://mlsanity.com/paper/2408.08071/) (2024)
- [Predictive Modeling in the Reservoir Kernel Motif Space](https://mlsanity.com/paper/2405.07045/) (2024)
- [Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics](https://mlsanity.com/paper/2507.06050/) (2025)
- [Structuring Multiple Simple Cycle Reservoirs with Particle Swarm Optimization](https://mlsanity.com/paper/2504.05347/) (2025)
- [HypER: Hyperbolic Echo State Networks for Capturing Stretch-and-Fold Dynamics in Chaotic Flows](https://mlsanity.com/paper/2508.18196/) (2025)
- [Symplectic Representation of Legendre Dynamics](https://mlsanity.com/paper/2512.19409/) (2025)
- [Explicit Interaction Architectures for Dynamical Learning: A Controlled Study of Structural Inductive Bias](https://mlsanity.com/paper/2606.19101/) (2026)

## Abstract (arXiv)

Reservoir computation models form a subclass of recurrent neural networks with fixed non-trainable input and dynamic coupling weights. Only the static readout from the state space (reservoir) is trainable, thus avoiding the known problems with propagation of gradient information backwards through time. Reservoir models have been successfully applied in a variety of tasks and were shown to be universal approximators of time-invariant fading memory dynamic filters under various settings. Simple cycle reservoirs (SCR) have been suggested as severely restricted reservoir architecture, with equal weight ring connectivity of the reservoir units and input-to-reservoir weights of binary nature with the same absolute value. Such architectures are well suited for hardware implementations without performance degradation in many practical tasks. In this contribution, we rigorously study the expressive power of SCR in the complex domain and show that they are capable of universal approximation of any unrestricted linear reservoir system (with continuous readout) and hence any time-invariant fading memory filter over uniformly bounded input streams.