Shallow Recurrent Decoder for Reduced Order Modeling of Plasma Dynamics

TL;DR

The paper tackles the computational bottleneck of high-fidelity ${\bf E}\times {\bf B}$ plasma simulations by introducing SHALLOW REcurrent Decoder (SHRED), a reduced-order modeling framework that encodes time-series sensor data with an LSTM and decodes it to a full high-dimensional state via a shallow network. Grounded in the separation-of-variables principle, SHRED can operate on compressive representations derived from a low-rank basis ${\bf V}^{(k)}$ obtained through randomized SVD, enabling reconstruction of 14 coupled plasma fields from as few as three sensors and providing neural-network roll-outs for forecasting. The approach is demonstrated on a Hall-thruster–representative 2D radial-azimuthal problem, achieving accurate reconstructions and credible forecasts while drastically reducing training and inference costs (e.g., through randomized SVD reducing preprocessing time). These results suggest SHRED as a viable path toward fast, data-driven digital twins for plasma devices, with potential integration of measurements and simulations to refine physical models. Notation: the full high-dimensional state lies in ${\bf x}_T\in\mathbb{R}^n$ and the sensor measurements are ${\bf y}_t= {\bf C}{\bf x}_t$, with the low-rank representation ${\bf X}^{(k)}={\bf U}^{(k)}{\boldsymbol\Sigma}^{(k)}{{\bf V}^{(k)}}^*$ used to train a compact SHRED model.

Abstract

Reduced order models are becoming increasingly important for rendering complex and multiscale spatio-temporal dynamics computationally tractable. The computational efficiency of such surrogate models is especially important for design, exhaustive exploration and physical understanding. Plasma simulations, in particular those applied to the study of ${\bf E}\times {\bf B}$ plasma discharges and technologies, such as Hall thrusters, require substantial computational resources in order to resolve the multidimentional dynamics that span across wide spatial and temporal scales. Although high-fidelity computational tools are available to simulate such systems over limited conditions and in highly simplified geometries, simulations of full-size systems and/or extensive parametric studies over many geometric configurations and under different physical conditions are computationally intractable with conventional numerical tools. Thus, scientific studies and industrially oriented modeling of plasma systems, including the important ${\bf E}\times {\bf B}$ technologies, stand to significantly benefit from reduced order modeling algorithms. We develop a model reduction scheme based upon a {\em Shallow REcurrent Decoder} (SHRED) architecture. The scheme uses a neural network for encoding limited sensor measurements in time (sequence-to-sequence encoding) to full state-space reconstructions via a decoder network. Based upon the theory of separation of variables, the SHRED architecture is capable of (i) reconstructing full spatio-temporal fields with as little as three point sensors, even the fields that are not measured with sensor feeds but that are in dynamic coupling with the measured field, and (ii) forecasting the future state of the system using neural network roll-outs from the trained time encoding model.

Figures (7)

• Figure 1: Architecture of the SHRED model for emulating plasma dynamics. A single field is measured, specifically ${\bf n}_e$, using sensor history (trajectory) data and the model is trained to map to the original data of the fourteen fields. In this case, the SHRED model is trained specifically to map to the compressive representation of the full plasma dynamics by mapping to the $r$-rank right singular values (${\bf V}^{(k)}$) of a given field computed by a randomized singular value decomposition ${\bf X}^{(k)} = {\bf U}^{(k)} {\bf \Sigma}^{(k)} {{\bf V}^{(k)}}^*$. Reconstruction of the $k$th field can be accomplished by projecting to the high-dimensional space using ${{\bf U}^{(k)}}^*$.
• Figure 2: First six principal components (SVD modes ${\bf u}_k$ for $k=1, 2, 3, 4, 5, 6$) of the ${\bf n}_e$ dynamics. The resolution of the simulation is $n_x=256 \times n_z=257$ in the $x-z$ plane for a total state space of each spatio-temporal field of $n= 65792$. Note that the horizontal axis represents the azimuthal direction ($z$) and the vertical axis represents the radial direction ($x$). The first 20 modes are retained for the training the model. The associated temporal dynamics are shown in Fig. \ref{['fig:svd2']}.
• Figure 3: (a) Singular value ($\sigma$) decay and percentage of variance (${100\sigma_j}/{\sum \sigma_k}$) captured by each mode of the randomized SVD decomposition. The first 20 modes are retained for SHRED training as these modes capture the dominant activity in the plasma. The time dynamics of the first 5 modes (${\bf v}_k$ for $k=1, 2, 3, 4, 5$) are illustrated in panels (b)-(f) as a function of time. The associated spatial modes are shown in Fig. \ref{['fig:svd1']}.
• Figure 4: Reconstruction versus truth for the fourteen spatio-temporal fields of the plasma dynamics. In this example, the ${\bf n}_e$ field is measured in three randomly locations. The sensor trajectory is used in SHRED to perform a reconstruction of all fourteen fields. The truth and reconstructions are shown for withheld test data at a time randomly selected. As is shown, three point sensor measurements are capable of accurate reconstructions. Note that the horizontal axis represents the azimuthal direction ($z$) and the vertical axis represents the radial direction ($x$).
• Figure 5: Time dynamics of the fourteen spatio-temporal fields of the plasma dynamics at a randomly selected spatial coordinate. The blue line is the ground truth and the orange line is the prediction of the temporal dynamics on test data.