Adaptive symplectic model order reduction of parametric particle-based Vlasov–Poisson equation

Abstract

High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov–Poisson equations. Since the problem’s nondissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov–Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost.

Learning the dynamics of a one-dimensional plasma model with graph neural networks

“…In order to provide stronger evidence of the GNS generalization capabilities, we showcase a broad range of known kinetic plasma processes that the simulator is able to recover. These examples, present in both the original sheet model benchmarks [37][38][39][40][41] and other kinetic codes benchmarks [2,3,9,42,50], aim to demonstrate the capability of the GNS to simulate collective behavior in accordance with known kinetic theory. An important point to stress is that the surrogate simulator was not explicitly trained to reproduce these effects.…”

“…To obtain computational speed-ups, there have been recent attempts to combine existing PIC codes with machine learning surrogate models. These efforts include approaches to accelerate [5] or fully replace [6,7] the field solver block, reduce the computational burden associated with the particle push and grid-particle/particle-grid interpolation [8,9], and the integration of surrogate models into advanced physics extensions [10]. In parallel, PIC simulations and machine learning algorithms have also been used to train fast surrogate models for plasma accelerator setups [11][12][13][14], to learn closures for fluid simulations [15], to model hybrid plasma representations [16], and to recover reduced plasma models [17].…”