Online learning of both state and dynamics using ensemble Kalman filters

Abstract

<p style='text-indent:20px;'>The reconstruction of the dynamics of an observed physical system as a surrogate model has been brought to the fore by recent advances in machine learning. To deal with partial and noisy observations in that endeavor, machine learning representations of the surrogate model can be used within a Bayesian data assimilation framework. However, these approaches require to consider long time series of observational data, meant to be assimilated all together. This paper investigates the possibility to learn both the dynamics and the state online, i.e. to update their estimates at any time, in particular when new observations are acquired. The estimation is based on the ensemble Kalman filter (EnKF) family of algorithms using a rather simple representation for the surrogate model and state augmentation. We consider the implication of learning dynamics online through (ⅰ) a global EnKF, (ⅰ) a local EnKF and (ⅲ) an iterative EnKF and we discuss in each case issues and algorithmic solutions. We then demonstrate numerically the efficiency and assess the accuracy of these methods using one-dimensional, one-scale and two-scale chaotic Lorenz models.</p>

Combining ensemble Kalman filter and reservoir computing to predict spatiotemporal chaotic systems from imperfect observations and models

“…We showed that our proposed approach works correctly. Brajard et al (2020) also made a CNN learn the dynamics from sparse observation data and successfully predict the dynamics of the L96 model. However, as mentioned in the introduction section, Brajard et al (2020) iterated the learning and data assimilation until they converge, because it replaced the model used in data assimilation with CNN.…”

“…Brajard et al (2020) also made a CNN learn the dynamics from sparse observation data and successfully predict the dynamics of the L96 model. However, as mentioned in the introduction section, Brajard et al (2020) iterated the learning and data assimilation until they converge, because it replaced the model used in data assimilation with CNN. Although their model-free method has an advantage that it was not affected by the process-based model's reproducibility of the phenomena, it can be computationally expensive since the number of iterations can be relatively large.…”

“…It can be a serious problem since the observation sparsity is often the case in geosciences and the NWP systems. Brajard et al (2020) pointed out this issue and successfully trained the convolutional neural network (CNN) with sparse observations, by combining it with EnKF. However, their method needs to iterate the data assimilation and training until the prediction accuracy of the trained model converges.…”

“…Although one can stop the iteration after a few time steps, training can be computationally expensive if one waits for convergence. Bocquet et al (2020) proposed a method to combine EnKF and machine learning methods to obtain both the state estimation and the surrogate model online. They showed successful results without using the process-based model at all.…”

Inferring the instability of a dynamical system from the skill of data assimilation exercises

“…Among those, we are currently considering how these results will change when performing DA for state and parameter estimation. In this context, a relevant recent study has shown how the minimum number of ensemble members, N * , will need to be increased to include as many members as the number of parameters to be estimated (Bocquet et al, 2020). By modifying its parameters, the model's instabilities properties will change too, potentially inducing a catastrophic change (a tipping point) of its long term behavior.…”