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.
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.
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.