# Infinite-dimensional reservoir computing

arXiv: https://arxiv.org/abs/2304.00490 · HTML: https://mlsanity.com/paper/2304.00490/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 73 references that are in the map.
- First posted on arXiv 2 Apr 2023.

## Builds on (most cited first, 30 of 34)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Temporal Information Processing on Noisy Quantum Computers](https://mlsanity.com/paper/2001.09498/) (2020)
- [Discrete-time signatures and randomness in reservoir computing](https://mlsanity.com/paper/2010.14615/) (2020)
- [Risk bounds for reservoir computing](https://mlsanity.com/paper/1910.13886/) (2019)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Learning strange attractors with reservoir systems](https://mlsanity.com/paper/2108.05024/) (2021)
- [Learning Nonlinear Input-Output Maps with Dissipative Quantum Systems](https://mlsanity.com/paper/1901.01653/) (2019)
- [Dynamical Systems as Temporal Feature Spaces](https://mlsanity.com/paper/1907.06382/) (2019)
- [Memory and forecasting capacities of nonlinear recurrent networks](https://mlsanity.com/paper/2004.11234/) (2020)
- [Dimension reduction in recurrent networks by canonicalization](https://mlsanity.com/paper/2007.12141/) (2020)
- [Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs](https://mlsanity.com/paper/2204.06255/) (2022)
- [Higher-Order Quantum Reservoir Computing](https://mlsanity.com/paper/2006.08999/) (2020)
- [Chaotic Hedging with Iterated Integrals and Neural Networks](https://mlsanity.com/paper/2209.10166/) (2022)

## Built on by (most cited first, 7 of 7)

- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [A Theoretical Analysis of the Test Error of Finite-Rank Kernel Ridge Regression](https://mlsanity.com/paper/2310.00987/) (2023)
- [Random Feature Representation Boosting](https://mlsanity.com/paper/2501.18283/) (2025)
- [RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks](https://mlsanity.com/paper/2411.06225/) (2024)
- [Fading memory and the convolution theorem](https://mlsanity.com/paper/2408.07386/) (2024)
- [Universality of reservoir systems with recurrent neural networks](https://mlsanity.com/paper/2403.01900/) (2024)
- [Vestibular reservoir computing](https://mlsanity.com/paper/2604.09943/) (2026)

## Abstract (arXiv)

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state-space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a fully implementable recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality.