Shallow Recurrent Decoder for Reduced Order Modeling of Plasma Dynamics

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.

Data-driven local operator finding for reduced-order modelling of plasma systems: II. Application to parametric dynamics

Real-world systems often exhibit dynamics influenced by various parameters, either inherent or externally controllable, necessitating models capable of reliably capturing these parametric behaviors. Plasma technologies exemplify such systems. For example, phenomena governing global dynamics in Hall thrusters (a spacecraft propulsion technology) vary with various parameters, such as the "self-sustained electric field". In this Part II, following on the introduction of our novel data-driven local operator finding algorithm, Phi Method, in Part I, we showcase the method's effectiveness in learning parametric dynamics to predict system behavior across unseen parameter spaces. We present two adaptations: the "parametric Phi Method" and the "ensemble Phi Method", which are demonstrated through 2D fluid-flow-past-a-cylinder and 1D Hall-thruster-plasma-discharge problems. Comparative evaluation against parametric OPT-DMD in the fluid case demonstrates superior predictive performance of the parametric Phi Method. Across both test cases, parametric and ensemble Phi Method reliably recover governing parametric PDEs and offer accurate predictions over test parameters. Ensemble ROM analysis underscores Phi Method's robust learning of dominant dynamic coefficients with high confidence.

Data-driven local operator finding for reduced-order modelling of plasma systems: I. Concept and verifications

Reduced-order plasma models that can efficiently predict plasma behavior across various settings and configurations are highly sought after yet elusive. The demand for such models has surged in the past decade due to their potential to facilitate scientific research and expedite the development of plasma technologies. In line with the advancements in computational power and data-driven methods, we introduce the "Phi Method" in this two-part article. Part I presents this novel algorithm, which employs constrained regression on a candidate term library informed by numerical discretization schemes to discover discretized systems of differential equations. We demonstrate Phi Method's efficacy in deriving reliable and robust reduced-order models (ROMs) for three test cases: the Lorenz attractor, flow past a cylinder, and a 1D Hall-thruster-representative plasma. Part II will delve into the method's application for parametric dynamics discovery. Our results show that ROMs derived from the Phi Method provide remarkably accurate predictions of systems' behavior, whether derived from steady-state or transient-state data. This underscores the method's potential for transforming plasma system modeling.

Dynamic Mode Decomposition for data-driven analysis and reduced-order modelling of ExB plasmas: I. Extraction of spatiotemporally coherent patterns

In this two-part article, we evaluate the utility and the generalizability of the Dynamic Mode Decomposition (DMD) algorithm for data-driven analysis and reduced-order modelling of plasma dynamics in cross-field ExB configurations. The DMD algorithm is an interpretable data-driven method that finds a best-fit linear model describing the time evolution of spatiotemporally coherent structures (patterns) in data. We have applied the DMD to extensive high-fidelity datasets generated using a particle-in-cell (PIC) code based on a cost-efficient reduced-order PIC scheme. In this part, we first provide an overview of the concept of DMD and its underpinning Proper Orthogonal and Singular Value Decomposition methods. Two of the main DMD variants are next introduced. We then present and discuss the results of the DMD application in terms of the identification and extraction of the dominant spatiotemporal modes from high-fidelity data over a range of simulation conditions. We demonstrate that the DMD variant based on variable projection optimization (OPT-DMD) outperforms the basic DMD method in identification of the modes underlying the data, leading to notably more reliable reconstruction of the ground-truth. Furthermore, we show in multiple test cases that the discrete frequency spectrum of OPT-DMD-extracted modes is consistent with the temporal spectrum from the Fast Fourier Transform of the data. This observation implies that the OPT-DMD augments the conventional spectral analyses by being able to uniquely reveal the spatial structure of the dominant modes in the frequency spectra, thus, yielding more accessible, comprehensive information on the spatiotemporal characteristics of the plasma phenomena.

Dynamic Mode Decomposition for data-driven analysis and reduced-order modelling of ExB plasmas: II. dynamics forecasting

In part I of the article, we demonstrated that a variant of the Dynamic Mode Decomposition (DMD) algorithm based on variable projection optimization, called Optimized DMD (OPT-DMD), enables a robust identification of the dominant spatiotemporally coherent modes underlying the data across various test cases representing different physical parameters in an ExB simulation configuration. As the OPT-DMD can be constrained to produce stable reduced-order models (ROMs) by construction, in this paper, we extend the application of the OPT-DMD and investigate the capabilities of the linear ROM from this algorithm toward forecasting in time of the plasma dynamics in configurations representative of the radial-azimuthal and axial-azimuthal cross-sections of a Hall thruster and over a range of simulation parameters in each test case. The predictive capacity of the OPT-DMD ROM is assessed primarily in terms of short-term dynamics forecast or, in other words, for large ratios of training-to-test data. However, the utility of the ROM for long-term dynamics forecasting is also presented for an example case in the radial-azimuthal configuration. The model's predictive performance is heterogeneous across various test cases. Nonetheless, a remarkable predictiveness is observed in the test cases that do not exhibit highly transient behaviors. Moreover, in all investigated cases, the error between the ground-truth and the reconstructed data from the OPT-DMD ROM remains bounded over time within both the training and the test window. As a result, despite its limitation in terms of generalized applicability to all plasma conditions, the OPT-DMD is proven as a reliable method to develop low computational cost and highly predictive data-driven reduced-order models in systems with a quasi-periodic global evolution of the plasma state.