# Online model learning with data-assimilated reservoir computers

arXiv: https://arxiv.org/abs/2504.16767 · HTML: https://mlsanity.com/paper/2504.16767/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 16 references that are in the map.
- First posted on arXiv 23 Apr 2025.

## Builds on (most cited first, 6 of 6)

- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Auto-Encoded Reservoir Computing for Turbulence Learning](https://mlsanity.com/paper/2012.10968/) (2020)
- [Stability analysis of chaotic systems in latent spaces](https://mlsanity.com/paper/2410.00480/) (2024)
- [Real-time thermoacoustic data assimilation](https://mlsanity.com/paper/2106.06409/) (2021)
- [Learning ergodic averages in chaotic systems](https://mlsanity.com/paper/2001.04027/) (2020)
- [Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data](https://mlsanity.com/paper/2404.19660/) (2024)

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

- [Data-assimilated model-informed reinforcement learning](https://mlsanity.com/paper/2506.01755/) (2025)
- [Efficient Adaptation of ROMs for Unsteady Flows Using Data Assimilation](https://mlsanity.com/paper/2602.23188/) (2026)

## Abstract (arXiv)

We propose an online learning framework for forecasting nonlinear spatio-temporal signals (fields). The method integrates (i) dimensionality reduction, here, a simple proper orthogonal decomposition (POD) projection; (ii) a generalized autoregressive model to forecast reduced dynamics, here, a reservoir computer; (iii) online adaptation to update the reservoir computer (the model), here, ensemble sequential data assimilation. We demonstrate the framework on a wake past a cylinder governed by the Navier-Stokes equations, exploring the assimilation of full flow fields (projected onto POD modes) and sparse sensors. Three scenarios are examined: a na"ive physical state estimation; a two-fold estimation of physical and reservoir states; and a three-fold estimation that also adjusts the model parameters. The two-fold strategy significantly improves ensemble convergence and reduces reconstruction error compared to the na"ive approach. The three-fold approach enables robust online training of partially-trained reservoir computers, overcoming limitations of a priori training. By unifying data-driven reduced order modelling with Bayesian data assimilation, this work opens new opportunities for scalable online model learning for nonlinear time series forecasting.