Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulations

TL;DR

The paper tackles the computational bottleneck of hybrid-Vlasov simulations by developing graph neural network surrogates that forecast near-Earth plasma states on a 2D+$3V$ plane. It presents both deterministic (Graph-FM) and probabilistic (Graph-EFM) emulators trained on a four-run Vlasiator dataset, incorporating a magnetic-divergence penalty and CRPS-based calibration to ensure physical plausibility and uncertainty quantification. The emulators achieve substantial speedups on GPUs (orders of magnitude faster per next step than the original solver) while faithfully reproducing large-scale magnetospheric structures like the bow shock and magnetotail, and provide calibrated ensemble forecasts, albeit with underdispersion (SSR ~ 0.2–0.3). The work releases open datasets and code, highlighting the potential of data-driven surrogates to enable real-time space-weather forecasting and uncertainty quantification, with clear paths toward 3D Extensions and full VDF emulation.

Abstract

Hybrid-Vlasov simulations resolve ion-kinetic effects for modeling the solar wind-magnetosphere interaction, but even 5D (2D + 3V) simulations are computationally expensive. We show that graph-based machine learning emulators can learn the spatiotemporal evolution of electromagnetic fields and lower order moments of ion velocity distribution in the near-Earth space environment from four 5D Vlasiator runs performed with identical steady solar wind conditions. The initial ion number density is systematically varied, while the grid spacing is held constant, to scan the ratio of the characteristic ion skin depth to the numerical grid size. Using a graph neural network architecture operating on the 2D spatial simulation grid comprising 670k cells, we demonstrate that both a deterministic forecasting model (Graph-FM) and a probabilistic ensemble forecasting model (Graph-EFM) based on a latent variable formulation are capable of producing accurate predictions of future plasma states. A divergence penalty is incorporated during training to encourage divergence-freeness in the magnetic fields and improve physical consistency. For the probabilistic model, a continuous ranked probability score objective is added to improve the calibration of the ensemble forecasts. When trained, the emulators achieve more than two orders of magnitude speedup in generating the next time step relative to the original simulation on a single GPU compared to 100 CPUs for the Vlasiator runs, while closely matching physical magnetospheric response of the different runs. These results demonstrate that machine learning offers a way to make hybrid-Vlasov simulation tractable for real-time use while providing forecast uncertainty.

Figures (8)

• Figure 1: Schematic overview of the Graph-EFM forecasting framework. The input simulation state (particle number density depicted here) is given to the model, which consists of two components: a latent map that converts the two most recent states into a distribution over a low-dimensional latent variable, and an encode–process–decode process that maps the input history and a sampled latent value to the next predicted state. The Graph-FM model used for comparison is only composed of the deterministic mapping from the two previous simulation steps to the following one. Finally, the predicted state can be concatenated to the previous step and given back as input, enabling multi-step rollouts.
• Figure 2: Example $\rho$ Vlasiator ground truth, Graph-EFM ensemble mean, and forecast uncertainty for each run at lead time $t=30\,\mathrm{s}$ for a forecast in the test set.
• Figure 3: Forecast performance on the test set indicated by RMSE, CRPS, and SSR for all deterministic (Graph-FM) and probabilistic (Graph-EFM) models across variables $B_x$, $E_x$, $v_x$, and $\rho$ for lead times 1--30 s.
• Figure 4: Effect of the magnetic divergence penalty. Shown are RMSE for $B_x$ and $B_z$ (left, center) and the mean absolute magnetic divergence over these fields (right) for Graph-FM and Graph-EFM models trained with different divergence-loss weights. The Vlasiator reference is included as a black dashed line. Moderate penalty weights reduce divergence without affecting RMSE, whereas excessively large weights can push $\nabla\cdot\mathbf{B}$ below the physical reference and increase the error of the corresponding magnetic field components.
• Figure 5: Normalized RMSE, CRPS, and SSR differences for Graph-EFM models using 2, 5, and 10 ensemble members. Values are reported relative to the 2-member ensemble (zero line). The y-axis has a symmetric logarithmic scale, that is linear between -0.2 and 0.2.