# Stochastic dynamics learning with state-space systems

arXiv: https://arxiv.org/abs/2508.07876 · HTML: https://mlsanity.com/paper/2508.07876/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 89 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 23 of 23)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [Time Series Data Augmentation for Deep Learning: A Survey](https://mlsanity.com/paper/2002.12478/) (2020)
- [Recurrent Neural Networks for Time Series Forecasting: Current Status and Future Directions](https://mlsanity.com/paper/1909.00590/) (2019)
- [Machine Learning for Synthetic Data Generation: A Review](https://mlsanity.com/paper/2302.04062/) (2023)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Physical reservoir computing -- An introductory perspective](https://mlsanity.com/paper/2005.00992/) (2020)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [On Particle Methods for Parameter Estimation in State-Space Models](https://mlsanity.com/paper/1412.8695/) (2014)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Combining Machine Learning with Knowledge-Based Modeling for Scalable Forecasting and Subgrid-Scale Closure of Large, Complex, Spatiotemporal Systems](https://mlsanity.com/paper/2002.05514/) (2020)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [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)
- [Dimension reduction in recurrent networks by canonicalization](https://mlsanity.com/paper/2007.12141/) (2020)
- [Optimal transport and Wasserstein distances for causal models](https://mlsanity.com/paper/2303.14085/) (2023)
- [Embedding Information onto a Dynamical System](https://mlsanity.com/paper/2105.10766/) (2021)
- [Transport in reservoir computing](https://mlsanity.com/paper/2209.07946/) (2022)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [Fading memory and the convolution theorem](https://mlsanity.com/paper/2408.07386/) (2024)

## Built on by (most cited first, 3 of 3)

- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)
- [Symplectic Representation of Legendre Dynamics](https://mlsanity.com/paper/2512.19409/) (2025)
- [Data-Specific Hyper-Parameter Design: A Paradigm Shift in Reservoir Computing](https://mlsanity.com/paper/2605.25221/) (2026)

## Abstract (arXiv)

This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish that fading memory and solution stability hold generically -- even in the absence of the ESP -- offering a robust explanation for the empirical success of RC models without strict contractivity conditions. In the stochastic case, we critically assess stochastic echo states, proposing a novel distributional perspective rooted in attractor dynamics on the space of probability distributions, which leads to a rich and coherent theory. Our results extend and generalize previous work on non-autonomous dynamical systems, offering new insights into causality, stability, and memory in RC models. This lays the groundwork for reliable generative modeling of temporal data in both deterministic and stochastic regimes.