Session PO07: Fundamental Plasma Physics: Computation and machine learning

An implicit, conservative, asymptotic-preserving electrostatic particle-in-cell algorithm for arbitrary magnetic fields

We propose a new electrostatic particle-in-cell algorithm able to use large timesteps compared to particle gyro-period in non-uniform external magnetic fields [1,2]. The algorithm extends earlier electrostatic fully implicit PIC implementations [3] with a new asymptotic- preserving (AP) particle-push scheme [4] that allows timesteps much larger than particle gyroperiods. In the large-timestep limit, the AP integrator preserves all the averaged particle drifts, while recovering particle full orbits with small timesteps. The scheme allows for a seamless, efficient treatment of particles in coexisting magnetized and unmagnetized regions, conserves energy and charge exactly, and does not spoil implicit solver performance. Non-uniform magnetic fields require modifications to the standard Crank-Nicolson orbit integrator [4,5] to capture all first-order drifts for timesteps much larger than the gyrofrequency. We demonstrate by numerical experiment that significant speedups are possible vs. the standard fully implicit electrostatic PIC algorithm without sacrificing solution quality and while preserving strict charge and energy conservation. 

[1] G. Chen and L. Chacón, J. Comput. Phys., 487 (2023)

[2] O. Koshkarov, L. Chacón, G. Chen, L. Ricketson, in preparation

[3] G. Chen, L. Chacón, and D. C. Barnes, J. Comput. Phys., 230 (2011)

[4] L. Ricketson and L. Chacón. J. Comput. Phys., 418 (2020)

[5] O. Koshkarov, L. Chacón, G. Chen, L. Ricketson, J. Comput. Phys., 459, 111146 (2022)   

Stability of energy-conserving PIC algorithms with smoother interpolations and more accurate Poisson solves

We have performed periodic, one-dimensional particle-in-cell (PIC) simulations to test explicit energy-conserving PIC algorithms with a variety of properties: continuous interpolated electric fields, continuous first derivatives of the interpolated electric field, Lagrangian-derived field solves, and second- and forth-order accurate field solves. We present numerical measurements of the stability, accuracy, and noise for each method, as well as comment on the generalizability to complex boundary conditions. In all cases, we find that the algorithm suffers from a cold-beam instability when v_thermal < v_drift and when v_drift < v_critical where v_critical is some algorithm-dependent critical velocity. Notably, all algorithms are stable to grid-heating (v_drift = 0) instabilities. This work confirms the semi-analytic work of Barnes and Chacon (2021), and extends their results by considering algorithms that have continuous first derivatives of the interpolated electric field and algorithms with a variety of field solves. We present an argument for the general failure of field solves to prevent cold-beam instabilities and derive v_critical analytically for some algorithms.   

Long-time-scale particle-in-cell simulations of magnetized plasmas

Electromagnetic particle-in-cell (PIC) simulations are key tools for modeling plasma dynamics in many fusion-relevant applications. Practical limitations on the time and computational resources that can be allocated to a given simulation, however, often limit their applicability to small spaciotemporal windows. Leveraging second-order, charge- and energy-conserving algorithms, our new code named TriForce aims to expand the spaciotemporal limits of PIC simulations. TriForce is a hybrid PIC/fluid code currently under development with an extensible and modular developer environment for multiphysics simulations. We present computational modeling details for long-time-scale simulations of three example systems: field-reversed configuration plasma, power flow in vacuum magnetically-insulated transmission lines and burn-wave propagation in magnetized, fusion-relevant plasma.   

Machine learning surrogate model for solving the Vlasov equation

The Vlasov equation provides a kinetic description of plasma dynamics in phase space. Solving the Vlasov equation is computationally expensive, often leading to the use of fluid models as alternatives. Fluid models describe macroscopic quantities like density, velocity, and pressure, while the Vlasov equation offers a microscopic perspective, capturing the detailed kinetic behavior of plasmas under the influence of electromagnetic fields. This allows for a more precise understanding of phenomena such as wave-particle interactions, instabilities, and nonlinear effects that are critical in plasma dynamics. The computational expense of solving the Vlasov equation and the challenges of traditional numerical methods in practical applications have motivated the exploration of alternative approaches that can balance accuracy and computational efficiency. Machine learning methods have the potential to overcome these challenges, and here we introduce the application of the Deep Operator Network (DeepONet) for solving the Vlasov equation. By training DeepONet on data generated from a Vlasov solver, it is able to output the precise solutions of the Vlasov equation, capturing the evolution of the plasma quantities such as density, pressure as well as electric field energy. This advancement holds significant potential for efficiently and accurately modeling plasma dynamics and advancing both theoretical and applied plasma physics research.   

Machine Learning Heat Flux Closure for Multi-Moment Fluid Modeling of Nonlinear Landau Damping

Nonlinear plasma physics problems are typically simulated through comprehensive modeling of phase space, incurring extreme computational costs. This has driven the development of multi-moment fluid models that integrates the Vlasov equation. However, a significant challenge remains in identifying a suitable fluid closure for these models. Recent advancements in physics-informed machine learning have sparked renewed interests in constructing accurate fluid closure terms. In this study, we present a novel approach that integrates kinetic physics from first principal Vlasov simulation data into a multi-moment fluid model through the heat flux closure term using the Fourier neural operator (FNO) —a specialized neural network architecture. For the first time, without resolving phase space dynamics, the newly developed fluid model accurately captures the nonlinear evolution of the Landau damping process, matching the fully kinetic simulation data precisely. This machine learning-assisted framework provides a computationally affordable method that surpasses previous fluid models in accurately modeling the kinetic evolution of complex plasma systems.   

Data-driven discovery of heat-flux closures for collisionless electrostatic plasma phenomena

Global modeling of multi-scale collisionless plasma phenomena is a long-standing computational challenge. Collisionless fluid models offer an approximate but more tractable alternative to fully kinetic modeling, calling for a systematic approach to constructing accurate closures that capture the essence of the kinetic physics. We employ data-driven methods [E.P. Alves and F. Fiuza 2022 Phys. Rev. Res. 4, 033192] based on the SINDy algorithm [S.L. Brunton et al. 2016 PNAS 113, 3932], to obtain a fluid closure in two-stream unstable and Landau-damped plasmas. More specifically, using OSIRIS particle-in-cell simulation data, we search for expressions for the heat flux in terms of lower order moments, which are optimally accurate at each given model complexity. Besides a local approximation of the Hammett-Perkins closure [P. Sharma et al. 2006 Astrophys. J. 637, 952; G.W. Hammett and F.W. Perkins 1990 Phys. Rev. Lett. 64, 3019], we find several additional closure terms - largely the same set in both the two-stream unstable and Landau-damped scenarios. Some terms are present only when wave propagation or beam asymmetry breaks isotropy. The generality of the found closures with respect to parameter variations is explored.   

A multiscale hybrid particle-Maxwellian Coulomb-Collision Algorithm for Hybrid Kinetic-Fluid Simulations

In many plasma systems, Coulomb collisions, the primary interactions between charged particles, often exhibit significant time-scale separation. Traditional Monte Carlo (MC) methods [1] have a timestep accuracy constraint ν∆t ≪ 1 to resolve the collision fre- quency (ν) effectively [2]. This constraint becomes particularly stringent in scenarios involving self-collisions in the presence of high-Z species or inter-species collisions with substantial mass disparities, rendering such simulations extremely expensive or impractical.

To overcome these challenges, we propose a multiscale hybrid particle-Maxwellian model for hybrid kinetic-fluid simulations. Specifically, we introduce a collisional al- gorithm that utilizes Maxwellians (i.e., isotropic Gaussians) [4] for both highly col- lisional kinetic species and fluid entities. We employ the Lemons method [3] for particle-Maxwellian collisions, enhanced with a more careful treatment of low-relative- speed particles. Additionally, we introduce a new scheme that extends the standard TA method for pairing particle-particle binary collisions to accommodate arbitrary particle weights without compromising conservation properties.

The method is strictly conservative and may significantly outperform standard MC methods, with orders of magnitude improvement in computational efficiency. We will substantiate the accuracy and performance of the proposed method through several examples of varying complexity, encompassing both relaxation and transport problems.

[1] Takizuka, T., Abe, H. Journal of computational physics, (1977) 25: 205-219.

[2] Dimits, A. M., Wang, C., Caflisch, R., Cohen, B. I., & Huang, Y. Journal of Computational Physics (2009) 228: 4881-4892.

[3] Lemons, D.S., Winske, D., Daughton, W. and Albright, B., Journal of Computa- tional Physics, (2009) 228(5), pp.1391-1403.

[4] EchimMM,LemaireJ,&Lie-SvendsenØ.Surveysingeophysics(2011)32:1-70.   

Data-driven discovery of sub-grid models for a turbulent electromotive force in magnetic reconnection

Sub-grid closures are often employed to capture the effects of small-scale, kinetic physics on larger scales, reducing the need to resolve large scale separations. While there is no general prescription for discovering such models, recent advances in machine learning techniques may offer mechanisms for discovering closures directly from data. We present an approach to sub-grid closure discovery in which physics-informed constraints are applied to shallow, fully convolutional neural networks (FCNs), implicitly revealing information about the physics needed for closure. We apply this framework to the nonlinear, multi-scale dynamics of coalescing magnetic islands in a low-beta, strongly magnetized pair plasma. We derive a simple analytic expression for a turbulent emf (A) describing the feedback of electron inertia scale physics on coarse-grained, MHD-scale fields and show via a systematic reduction of the FCN receptive field that the dynamics encoded in A are accurately captured using only spatially local patches of the coarse-grained fields as inputs. To better understand the validity and generalizability of our results, we evaluate sensitivity to the specific coarse-graining procedure and assess performance on a variety of physically motivated metrics for multiple simulations at different scale-separations.   

A generative artificial intelligence surrogate model of plasma turbulence

Turbulent transport plays a key role in confinement degradation limiting the performance of current and future fusion devices. Modelling turbulent transport requires long- time simulations which are limited by the computational resources available. One way to overcome this shortcoming is by using surrogate models that are computationally cheaper to evaluate. In this presentation we apply generative artificial intelligence (AI) methods to construct a surrogate model of plasma edge turbulence described by the HW (Hasegawa-Wakatani) model and use the model to perform fast, long-time turbulent transport computations.

The proposed GAIT (Generative Artificial Intelligence Turbulence) model is based on the combination of a convolutional variational autoencoder and a deep neural network (DNN). A convolutional network is used to encode snapshots of computed HW turbulence states into a reduced latent space, and a DNN is trained to reproduce the time evolution of turbulence in the latent space. Once the autoencoder is trained, new turbulence states are obtained by decoding the latent space dynamics generated by the DNN.

To evaluate the model we use Eulerian and Lagrangian metrics. Good agreement is found between the GAIT and the HW models in the spatial and temporal Eulerian turbulence Fourier spectra. In the Lagrangian setting, the statistical moments and probability distribution of particle displacements are compared, and agreement is found in the effective diffusivity in the GAIT and the HW models.   

Modelling Rapid Target Heating through the Discontinuous Galerkin and Material Point Methods

We present results from our efforts to model the plume generated by the rapid heating of an aluminum target via a time-dependent heat source that emulates an electron beam.

The plasma plumes generated by such extreme heating processes can impact experiments, making an accurate prediction of their behavior important for both design and analysis. Experiments were performed using rapid pulses of about 10¹⁵ electrons to heat targets to temperatures in excess of 1 eV, with shadowgraph and interferometer measurements providing snapshots of the plume’s temperature and electron density.

We approximate this high mach number experiment in two ways. The first is using the Discontinuous Galerkin method to simulate ideal-gas hydrodynamics to model plume evolution, specifically using a slope limiter and positivity preserver to model the shocks in the simulations. Next is using the material point method with a linear elastic and ideal gas model coupled via a phase change. Initial simulations show promise in DG’s capability to model the rapid expansion of the plume, while MPM simulations are able to handle the solid mechanics of the target well.

Research presented is supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory.   

Modeling Plasma Plumes from Rapid Target Heating with Differentiable Smoothed Particle Hydrodynamics

We present a fully differentiable Smoothed Particle Hydrodynamics (SPH) simulator for modeling the hydrodynamic expansion of plasma plumes resulting from rapid target heating. Many formulations of SPH have been developed for modeling compressible flows involving plasma, such as those seen in astrophysics and engineering applications. However, the parameters of these SPH formulations are usually adjusted by trial and error. In this work, we leverage differentiable programming to automatically fine-tune SPH parameters to experimental data. We embed Neural Networks withing the SPH framework for estimating unknown functions, such as smoothing kernels and equations of state. 

The approach is first validated and analyzed for the 1d Sod shock and 2D Taylor-Sedov blast wave problems, i.e. learn-able and parameterized compressible SPH formulations are fit to the analytical solutions using gradient-based optimization. Next, we extend the SPH formulation to model the spatio-temporal evolution of plasma plumes generated from rapid thin-target heating by 20 MeV electron beams. The generation of plasma plumes during such heating processes can significantly impact various high-energy physics experiments, making accurate prediction of their behavior crucial for experimental design and analysis. The plasma plumes of interest evolve in vacuum, making the SPH framework a natural choice for modeling at the coarse grained scales of interest. Furthermore, the complex physics involved motivates the use of machine learning to fill in the gaps with exploring new parameterized terms within SPH. We employ experimental data to fine tune the SPH models, using pulsed bunches of approximately $10^{15}$ electrons to heat a range of thin targets to temperatures above 1 eV. The shadowgraph and interferometer measurements provide spatially resolved snapshots of the expanding plume's electron density and temperature distributions at discrete time points.