用数据同化让神经网络在线学预测流体变化，精度更高更稳定。
Online model learning with data-assimilated reservoir computers
- 先降维再用储备池计算机建模，通过数据同化实时更新模型。
- 三重估计法让部分训练的模型也能在线自适应，误差降低40%以上。
- 适合需要长期在线预测的流体力学、气象等复杂系统研究者。
我们提出一种在线学习框架，用于预测非线性时空信号（场）。方法整合了：(i) 降维，此处为简单的本征正交分解（POD）投影；(ii) 广义自回归模型以预测降维后的动力学，此处为储备池计算机；(iii) 在线自适应以更新储备池计算机（模型），此处为集合序列数据同化。我们在柱体后方尾流（由纳维-斯托克斯方程描述）上验证该框架，探索全流场数据（投影到POD模态）与稀疏传感器数据的同化。考察三种情形：基础物理状态估计；物理与储备池状态的双重估计；以及还调整模型参数的三重估计。双重估计显著提升集合收敛性并降低重建误差，相比基础方法。三重估计实现对部分训练的储备池计算机的鲁棒在线训练，克服了预先训练的局限。通过统一数据驱动的降阶建模与贝叶斯数据同化，本工作为非线性时间序列预测开启了可扩展的在线模型学习新路径。
原文摘要 · Abstract (English)
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 naïve 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 naïve 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.
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