State Estimation Using Sparse DEIM and Recurrent Neural Networks

TL;DR

An equation-free S-DEIM framework that estimates the optimal kernel vector from sparse observational time series using recurrent neural networks (RNNs) is introduced and it is shown that the recurrent architecture is necessary since the kernel vector cannot be estimated from instantaneous observations.

Abstract

Sparse Discrete Empirical Interpolation Method (S-DEIM) was recently proposed for state estimation in dynamical systems when only a sparse subset of the state variables can be observed. The S-DEIM estimate involves a kernel vector whose optimal value is inferred through a data assimilation algorithm. This data assimilation step suffers from two drawbacks: (i) It requires the knowledge of the governing equations of the dynamical system, and (ii) It is not generally guaranteed to converge to the optimal kernel vector. To address these issues, here we introduce an equation-free S-DEIM framework that estimates the optimal kernel vector from sparse observational time series using recurrent neural networks (RNNs). We show that the recurrent architecture is necessary since the kernel vector cannot be estimated from instantaneous observations. But RNNs, which incorporate the past history of the observations in the learning process, lead to nearly optimal estimations. We demonstrate the efficacy of our method on three numerical examples with increasing degree of spatiotemporal complexity: a conceptual model of atmospheric flow known as the Lorenz-96 system, the Kuramoto-Sivashinsky equation, and the Rayleigh-Benard convection. In each case, the resulting S-DEIM estimates are satisfactory even when a relatively simple RNN architecture, namely the reservoir computing network, is used. More specifically, our RNN-based S-DEIM state estimations reduce the relative error between 42% and 58% when compared to Q-DEIM which ignores the kernel vector by setting it equal to zero.

Figures (11)

• Figure 1: State space geometry of DEIM estimation with $r<m$. The truncated basis $\{\pmb\phi_1,\cdots,\pmb\phi_m\}$ spans the subspace $\mathcal{R}[\Phi_m]\subset\mathbb R^N$. The variables denote the true state $\mathbf u$, the best fit $\hat{\mathbf u}$, DEIM estimate $\widetilde{\mathbf u}(\mathbf 0)$, and the optimal S-DEIM estimate $\widetilde{\mathbf u}(\hat{\mathbf z})$. The subspace $\mathcal{R}[\Phi_m]$ admits the orthogonal decomposition $\mathcal{R}[\Phi_m]=\mathcal{R}[(S_r^\top\Phi_m)^+]\oplus \Phi\mathcal{N}[S_r^\top\Phi_m]$.
• Figure 2: Pipeline of the proposed framework. In the offline stage, the POD modes and optimal sensor locations are computed and the RNN is trained. In the real-time reconstruction, the pretrained network is used to estimate the state from observational data. The observational time series from the training set is denoted by $\mathcal{Y}_{tr}(t)$.
• Figure 3: Architecture of the reservoir computing network.
• Figure 4: State estimation for Lorenz-96 system. The reconstructions are obtained from $r=10$ sensors and $m=20$ modes
• Figure 5: Relative errors for Lorenz-96 system. (a) Relative error as a function of time. The legend reports the mean relative error for each method. (b) A close-up view showing the first 10 time units.