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This paper introduces a neural score-based transport modeling (SBTM) approach to estimate the velocity score function within a particle method for simulating the Vlasov-Maxwell-Landau (VML) system, a key challenge in plasma modeling. SBTM replaces the traditional blob method, offering a linear cost ($O(n)$) compared to the blob method's quadratic cost ($O(n^2)$). The method is proven to preserve momentum and kinetic energy, dissipate entropy, and demonstrates superior accuracy and efficiency compared to the blob method on canonical plasma benchmarks, achieving correct long-time relaxation to Maxwellian equilibrium.
Neural score-based transport slashes the computational cost of simulating complex plasma kinetics while simultaneously boosting accuracy and ensuring correct long-time behavior.
Plasma modeling is central to the design of nuclear fusion reactors, yet simulating collisional plasma kinetics from first principles remains a formidable computational challenge: the Vlasov-Maxwell-Landau (VML) system describes six-dimensional phase-space transport under self-consistent electromagnetic fields together with the nonlinear, nonlocal Landau collision operator. A recent deterministic particle method for the full VML system estimates the velocity score function via the blob method, a kernel-based approximation with $O(n^2)$ cost. In this work, we replace the blob score estimator with score-based transport modeling (SBTM), in which a neural network is trained on-the-fly via implicit score matching at $O(n)$ cost. We prove that the approximated collision operator preserves momentum and kinetic energy, and dissipates an estimated entropy. We also characterize the unique global steady state of the VML system and its electrostatic reduction, providing the ground truth for numerical validation. On three canonical benchmarks -- Landau damping, two-stream instability, and Weibel instability -- SBTM is more accurate than the blob method, achieves correct long-time relaxation to Maxwellian equilibrium where the blob method fails, and delivers $50\%$ faster runtime with $4\times$ lower peak memory.
