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.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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