Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches

Abstract

Plasmas are highly nonlinear and multi-scale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduced models hold promise for understanding key physical mechanisms, efficient computation, and real-time optimization and control. Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, and data-driven models, obtained by modern machine learning and system identification, can furnish this gap in the lower levels of the model hierarchy. This work develops a reduced-order modeling framework for compressible plasmas, leveraging decades of progress in projection-based and data-driven modeling of fluids. We begin by formalizing projection-based model reduction for nonlinear MHD systems. To avoid separate modal decompositions for the magnetic, velocity, and pressure fields, we introduce an energy inner product to synthesize all of the fields into a dimensionally-consistent, reduced-order basis. Next, we obtain an analytic model by Galerkin projection of the Hall-MHD equations onto these modes. We illustrate how global conservation laws constrain the model parameters, revealing symmetries that can be enforced in data-driven models, directly connecting these models to the underlying physics. We demonstrate the effectiveness of this approach on data from high-fidelity numerical simulations of a 3D spheromak experiment. This manuscript builds a bridge to the extensive Galerkin literature in fluid mechanics, and facilitates future principled development of projection-based and data-driven models for plasmas.

References in corpus (19)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Machine learning accelerated computational fluid dynamics
• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• Shell Models of Magnetohydrodynamic Turbulence
• Formulating turbulence closures using sparse regression with embedded form invariance
• Inferring the dynamics of underdamped stochastic systems
• Sparse Identification of Nonlinear Dynamical Systems via Reweighted ${\ell }_1$-regularized Least Squares
• Nonlinear stochastic modeling with Langevin regression
• Data-driven discovery of reduced plasma physics models from fully-kinetic simulations
• Characterizing Magnetized Plasmas with Dynamic Mode Decomposition
• Lagrangian Neural Networks
• Hamiltonian Formalism of Extended Magnetohydrodynamics
• Sparse nonlinear models of chaotic electroconvection
• PySINDy: A Python package for the Sparse Identification of Nonlinear Dynamics from Data
• Learning Symbolic Physics with Graph Networks
• Sparse Identification of Slow Timescale Dynamics
• The use of the Biorthogonal Decomposition for the identification of zonal flows at TJ-II
• Advanced modeling for the HIT-SI Experiment
• Data-Driven Closure of Projection-Based Reduced Order Models for Unsteady Compressible Flows

Cited by in corpus (23)

• Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control
• PySINDy: A comprehensive Python package for robust sparse system identification
• On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues
• 2022 Review of Data-Driven Plasma Science
• Data-driven discovery of reduced plasma physics models from fully-kinetic simulations
• Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES
• Sparse nonlinear models of chaotic electroconvection
• Detection and Prediction of Equilibrium States in Kinetic Plasma Simulations via Mode Tracking using Reduced-Order Dynamic Mode Decomposition
• Permanent magnet optimization for stellarators as sparse regression
• In Search of a Data Driven Symbolic Multi-Fluid 10-Moment Model Closure
• Greedy permanent magnet optimization
• Data-driven Discovery of Delay Differential Equations with Discrete Delays
• Data-driven discovery of a heat flux closure for electrostatic plasma phenomena
• Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator
• Influence of initial conditions on data-driven model identification and information entropy for ideal mhd problems
• Promoting global stability in data-driven models of quadratic nonlinear dynamics
• Accelerating Particle-in-Cell Kinetic Plasma Simulations via Reduced-Order Modeling of Space-Charge Dynamics using Dynamic Mode Decomposition
• Energy-Preserving Reduced Operator Inference for Efficient Design and Control
• Extending the trapping theorem to provide local stability guarantees for quadratically nonlinear models
• Machine Learning in Viscoelastic Fluids via Energy-Based Kernel Embedding
• Embedding physical symmetries into machine-learned reduced plasma physics models via data augmentation
• An efficient algorithm for simulating ensembles of parameterized MHD flow problems
• Interpretable neural network system identification method for two families of second-order systems based on characteristic curves