Local Reduced-Order Modeling for Electrostatic Plasmas by Physics-Informed Solution Manifold Decomposition

TL;DR

This work develops reduced-order models for the collisionless electrostatic $1D1V$ Vlasov–Poisson system to enable fast multi-query simulations. It combines POD-based projection with a tensorial update for the nonlinear term and introduces temporally local ROMs formed via time- and energy-based solution-manifold indicators. The approach yields substantial speed-ups (up to ~$89\x$) while preserving accuracy (errors often within a few percent) across both reproduction and parametric scenarios, with EW-ROM frequently outperforming TW-ROM and single-ROM baselines. The results demonstrate the effectiveness of solution-manifold decomposition and tensorial nonlinear-term evaluation for advection-dominated kinetic problems, and point to extensions to higher dimensions and electromagnetics. Overall, the framework offers practical acceleration for parametric plasma simulations and establishes a pathway for scalable ROMs in kinetic models.

Abstract

Despite advancements in high-performance computing and modern numerical algorithms, computational cost remains prohibitive for multi-query kinetic plasma simulations. In this work, we develop data-driven reduced-order models (ROMs) for collisionless electrostatic plasma dynamics, based on the kinetic Vlasov-Poisson equation. Our ROM approach projects the equation onto a linear subspace defined by the proper orthogonal decomposition (POD) modes. We introduce an efficient tensorial method to update the nonlinear term using a precomputed third-order tensor. We capture multiscale behavior with a minimal number of POD modes by decomposing the solution manifold into multiple time windows and creating temporally local ROMs. We consider two strategies for decomposition: one based on the physical time and the other based on the electric field energy. Applied to the 1D1V Vlasov-Poisson simulations, that is, prescribed E-field, Landau damping, and two-stream instability, we demonstrate that our ROMs accurately capture the total energy of the system both for parametric and time extrapolation cases. The temporally local ROMs are more efficient and accurate than the single ROM. In addition, in the two-stream instability case, we show that the energy-windowing reduced-order model (EW-ROM) is more efficient and accurate than the time-windowing reduced-order model (TW-ROM). With the tensorial approach, EW-ROM solves the equation approximately 90 times faster than Eulerian simulations while maintaining a maximum relative error of 7.5% for the training data and 11% for the testing data.

Figures (19)

• Figure 1: Two-stream instability. The behavior of the FOM electric field energy $\int E^2~dx$ in time for parameters $\bm p = (\alpha, v_T) =(0.0025,0.08)$, $(0.0025,0.1)$, $(0.001,0.08)$, and $(0.001,0.1)$.
• Figure 2: An illustrative example to explain the mechanism of decomposition of solution manifold using the physical time and the electric field energy as indicator. The samples in each group are surrounded by a box with the same color, namely dark gray, green, green, pink, and orange. Groups classified using the electric field energy have strong linear dependence.
• Figure 3: Prescribed electric field case. The behavior of $1-\sum^{n_f}_{i=1} \sigma_i /\sum^K_{i=1} \sigma_i$ as a function of the reduced space dimension $n_f$.
• Figure 4: Prescribed electric field case. The behavior of the relative error in the distribution field at the final time $t_f$, $\epsilon_{f,t_f}$ with respect to energy missing ratio $\delta_\sigma$.
• Figure 5: Prescribed electric field case. The FOM and ROM solutions at time $t=130$ with energy missing ratio $\delta_\sigma = 10^{-2}$.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3