# Reservoir Computing Universality With Stochastic Inputs

arXiv: https://arxiv.org/abs/1807.02621 · HTML: https://mlsanity.com/paper/1807.02621/

## Key facts

- Cited by 48 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 7 Jul 2018.

## Built on by (most cited first, 30 of 48)

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Optimal Stopping via Randomized Neural Networks](https://mlsanity.com/paper/2104.13669/) (2021)
- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [A Survey on Reservoir Computing and its Interdisciplinary Applications Beyond Traditional Machine Learning](https://mlsanity.com/paper/2307.15092/) (2023)
- [Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing](https://mlsanity.com/paper/2108.04074/) (2021)
- [Learning strange attractors with reservoir systems](https://mlsanity.com/paper/2108.05024/) (2021)
- [A Brief Survey on the Approximation Theory for Sequence Modelling](https://mlsanity.com/paper/2302.13752/) (2023)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Learning unseen coexisting attractors](https://mlsanity.com/paper/2207.14133/) (2022)
- [Catch-22s of reservoir computing](https://mlsanity.com/paper/2210.10211/) (2022)
- [Controlling Chaos Using Edge Computing Hardware](https://mlsanity.com/paper/2406.12876/) (2024)
- [Inverse Approximation Theory for Nonlinear Recurrent Neural Networks](https://mlsanity.com/paper/2305.19190/) (2023)
- [Infinite-dimensional next-generation reservoir computing](https://mlsanity.com/paper/2412.09800/) (2024)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Universality and approximation bounds for echo state networks with random weights](https://mlsanity.com/paper/2206.05669/) (2022)
- [Memory of recurrent networks: Do we compute it right?](https://mlsanity.com/paper/2305.01457/) (2023)
- [Positional Encoding Helps Recurrent Neural Networks Handle a Large Vocabulary](https://mlsanity.com/paper/2402.00236/) (2024)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [Stochastic Reservoir Computers](https://mlsanity.com/paper/2405.12382/) (2024)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [Nonlinear Autoregression with Convergent Dynamics on Novel Computational Platforms](https://mlsanity.com/paper/2108.08001/) (2021)
- [Master memory function for delay-based reservoir computers with single-variable dynamics](https://mlsanity.com/paper/2108.12643/) (2021)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [Universality of Real Minimal Complexity Reservoir](https://mlsanity.com/paper/2408.08071/) (2024)
- [RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks](https://mlsanity.com/paper/2411.06225/) (2024)
- [Asymptotic evaluation of the information processing capacity in reservoir computing](https://mlsanity.com/paper/2502.15769/) (2025)
- [Effect of temporal resolution on the reproduction of chaotic dynamics via reservoir computing](https://mlsanity.com/paper/2302.10761/) (2023)
- [Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems](https://mlsanity.com/paper/2305.00669/) (2023)

## Abstract (arXiv)

The universal approximation properties with respect to L p -type criteria of three important families of reservoir computers with stochastic discrete-time semi-infinite inputs is shown. First, it is proved that linear reservoir systems with either polynomial or neural network readout maps are universal. More importantly, it is proved that the same property holds for two families with linear readouts, namely, trigonometric state-affine systems and echo state networks, which are the most widely used reservoir systems in applications. The linearity in the readouts is a key feature in supervised machine learning applications. It guarantees that these systems can be used in high-dimensional situations and in the presence of large datasets. The L p  criteria used in this paper allow the formulation of universality results that do not necessarily impose almost sure uniform boundedness in the inputs or the fading memory property in the filter that needs to be approximated.