# Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems

arXiv: https://arxiv.org/abs/2305.00669 · HTML: https://mlsanity.com/paper/2305.00669/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 71 references that are in the map.
- First posted on arXiv 1 May 2023.

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Using Machine Learning to Anticipate Tipping Points and Extrapolate to Post-Tipping Dynamics of Non-Stationary Dynamical Systems](https://mlsanity.com/paper/2207.00521/) (2022)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [Bayesian optimization of hyper-parameters in reservoir computing](https://mlsanity.com/paper/1611.05193/) (2016)
- [Learning unseen coexisting attractors](https://mlsanity.com/paper/2207.14133/) (2022)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)

## Built on by (most cited first, 2 of 2)

- [Learning noise-induced transitions by multi-scaling reservoir computing](https://mlsanity.com/paper/2309.05413/) (2023)
- [Stochastic parameter reduced-order model based on hybrid machine learning approaches](https://mlsanity.com/paper/2403.17032/) (2024)

## Abstract (arXiv)

The prediction of stochastic dynamical systems and the capture of dynamical behaviors are profound problems. In this article, we propose a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue, which mimics error modeling to improve traditional Reservoir Computing performance and integrates the virtues of both approaches. With few assumptions about the underlying stochastic dynamical systems, this model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. We verify the effectiveness of the proposed framework in several experiments, including the stochastic Van der Pal oscillator, El Ni~no-Southern Oscillation simplified model, and stochastic Lorenz system. These experiments consist of Markov/non-Markov and stationary/non-stationary stochastic processes which are defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. Additionally, we explore the noise-induced tipping phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and replication of the strange attractor.