# Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing

arXiv: https://arxiv.org/abs/2108.04074 · HTML: https://mlsanity.com/paper/2108.04074/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 21 references that are in the map.
- First posted on arXiv 6 Aug 2021.

## Builds on (most cited first, 3 of 3)

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)

## Built on by (most cited first, 21 of 21)

- [Learning unseen coexisting attractors](https://mlsanity.com/paper/2207.14133/) (2022)
- [Catch-22s of reservoir computing](https://mlsanity.com/paper/2210.10211/) (2022)
- [Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction](https://mlsanity.com/paper/2401.00885/) (2023)
- [Exploring the limits of multifunctionality across different reservoir computers](https://mlsanity.com/paper/2205.11375/) (2022)
- [Chaotic attractor reconstruction using small reservoirs -- the influence of topology](https://mlsanity.com/paper/2402.16888/) (2024)
- [Continual Learning of Dynamical Systems with Competitive Federated Reservoir Computing](https://mlsanity.com/paper/2206.13336/) (2022)
- [Stabilizing Machine Learning Prediction of Dynamics: Noise and Noise-inspired Regularization](https://mlsanity.com/paper/2211.05262/) (2022)
- [How more data can hurt: Instability and regularization in next-generation reservoir computing](https://mlsanity.com/paper/2407.08641/) (2024)
- [Master memory function for delay-based reservoir computers with single-variable dynamics](https://mlsanity.com/paper/2108.12643/) (2021)
- [Inferring Attracting Basins of Power System with Machine Learning](https://mlsanity.com/paper/2305.14374/) (2023)
- [Confabulation dynamics in a reservoir computer: Filling in the gaps with untrained attractors](https://mlsanity.com/paper/2505.04792/) (2025)
- [Learn one size to infer all: Exploiting translational symmetries in delay-dynamical and spatio-temporal systems using scalable neural networks](https://mlsanity.com/paper/2111.03706/) (2021)
- [Effect of temporal resolution on the reproduction of chaotic dynamics via reservoir computing](https://mlsanity.com/paper/2302.10761/) (2023)
- [Learning unidirectional coupling using echo-state network](https://mlsanity.com/paper/2303.13562/) (2023)
- [Learning Beyond Experience: Generalizing to Unseen State Space with Reservoir Computing](https://mlsanity.com/paper/2506.05292/) (2025)
- [Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics](https://mlsanity.com/paper/2507.06050/) (2025)
- [Multifunctional physical reservoir computing in soft tensegrity robots](https://mlsanity.com/paper/2507.21496/) (2025)
- [Thermodynamic limit in learning period three](https://mlsanity.com/paper/2405.08825/) (2024)
- [Unsupervised Learning in Echo State Networks for Input Reconstruction](https://mlsanity.com/paper/2501.11409/) (2025)
- [Denoising and Reconstruction of Nonlinear Dynamics using Truncated Reservoir Computing](https://mlsanity.com/paper/2504.13355/) (2025)
- [Inference of Unknown Dynamical Components Using Next Generation Reservoir Computing: From Chaotic Systems to Climate Data](https://mlsanity.com/paper/2609.24754/) (2026)

## Abstract (arXiv)

Reservoir computers are powerful tools for chaotic time series prediction. They can be trained to approximate phase space flows and can thus both predict future values to a high accuracy, as well as reconstruct the general properties of a chaotic attractor without requiring a model. In this work, we show that the ability to learn the dynamics of a complex system can be extended to systems with co-existing attractors, here a 4-dimensional extension of the well-known Lorenz chaotic system. We demonstrate that a reservoir computer can infer entirely unexplored parts of the phase space: a properly trained reservoir computer can predict the existence of attractors that were never approached during training and therefore are labelled as unseen. We provide examples where attractor inference is achieved after training solely on a single noisy trajectory.