math.NAMar 26, 2026

A Neural Score-Based Particle Method for the Vlasov-Maxwell-Landau System

Authors: Vasily Ilin, Jingwei Hu

Organizations: Department of Mathematics, University of Washington, Seattle, WA, USA. · Department of Applied Mathematics, University of Washington, Seattle, WA, USA.

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Abstract

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(n2)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)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%50\% faster runtime with 4×4\times lower peak memory.

Figures & tables

Figure 1: Landau damping at ν=0.4\nu=0.4 : estimated entropy production and total energy across particle counts. SBTM (red) dissipates estimated entropy consistently and maintains near-constant total energy. The blob method (blue) overestimates estimated entropy production and exhibits energy drift.
Figure 2: Landau damping: L2L^{2} norm of the electric field over time at ν=0.4\nu=0.4 . The linear theory damping rate is shown for reference (at ν=0.4\nu=0.4 it is not expected to be accurate). The blob method’s damping rate varies with nn (compare (a), (c), and (e)), converging toward the SBTM rate. The SBTM rate is consistent across all particle counts (b), (d), (f).
Figure 3: Landau damping: velocity-space distributions at n=106n=10^{6} . Rows 1–2: ν=0.4\nu=0.4 (heatmaps and v2v_{2} -marginal). Rows 3–4: ν=1.0\nu=1.0 . SBTM maintains smooth Gaussian tails; the blob method shows unphysical sharp cutoffs at ∣v2∣≈3|v_{2}|\approx 3 .
Figure 4: Two-stream instability: energy and estimated entropy diagnostics at ν=0.32\nu=0.32 . Blob (blue) vs SBTM (red) at n=5×105n=5\times 10^{5} (solid), 10610^{6} (dashed), 3×1063\times 10^{6} (dotted).
Figure 5: Two-stream instability: (x,v1)(x,v_{1}) phase space at ν=0.32\nu=0.32 across particle counts. SBTM achieves full vortex dissipation at all particle counts, while the blob method requires n=3×106n=3\times 10^{6} to approach the same level of dissipation.
Figure 6: Two-stream instability: score quiver plots at ν=0.24\nu=0.24 , n=106n=10^{6} . Top: initial time (true score in red; MSE in legend). Bottom: final time. SBTM produces smooth, physically consistent scores; the blob method has high MSE, particularly in low-density tail regions.
Figure 7: Two-stream instability: collision force at ν=0.24\nu=0.24 , n=5×105n=5\times 10^{5} . Top two rows: blob (noisy, incoherent fields that worsen as fine structure develops). Bottom two rows: SBTM (smooth, coherent fields at all times).
ν\nuBlobSBTM
2×10−42\times 10^{-4}0.3250.265
4×10−44\times 10^{-4}0.1340.053
8×10−48\times 10^{-4}0.0700.020
Table 1: Weibel instability: L2L^{2} distance of the v2v_{2} -marginal to N(0,T∞)\mathcal{N}(0,T_{\infty}) at tfinal=125t_{\mathrm{final}}=125 ( n=106n=10^{6} , dv=3d_{v}=3 ). Lower is better. SBTM relaxes closer to the Maxwellian steady state across all ν\nu .
Figure 8: Weibel instability: v2v_{2} -marginal density evolution at ν=8×10−4\nu=8\times 10^{-4} . Top: log scale; bottom: linear scale. The dashed black curve is a Gaussian/Maxwellian. SBTM converges to it with smooth Gaussian tails, while the blob method exhibits sharp, unphysical cutoffs at v2≈±0.55v_{2}\approx\pm 0.55 .
Figure 9: Weibel instability: (v1,v2)(v_{1},v_{2}) -marginal density. (a)–(d): phase space snapshots at ν=10−4\nu=10^{-4} and ν=8×10−4\nu=8\times 10^{-4} . (e)–(h): (v1,v2)(v_{1},v_{2}) density slices at x=0x=0 . At ν=0\nu=0 (e, f), both methods are identical. At ν=10−4\nu=10^{-4} (g, h), the blob method over-smooths while SBTM preserves fine structures.
Figure 10: Weibel instability: score quiver plots at ν=10−4\nu=10^{-4} . Top: initial time (true score in red; MSE in legend). Bottom: final time (true score unavailable). SBTM produces smooth, physically consistent scores; the blob method has high MSE at t=0t=0 and noisy scores at t=tfinalt=t_{\mathrm{final}} .
Figure 11: Weibel instability: estimated entropy production for blob (blue) and SBTM (red). Left: across collision frequencies; SBTM correctly decays to zero while the blob method plateaus. Right: across particle counts at ν=8×10−4\nu=8\times 10^{-4} ; the blob plateau decreases with nn , converging toward SBTM.
Figure 12: Weibel instability, dv=2d_{v}=2 : (v1,v2)(v_{1},v_{2}) -marginal density across collision frequencies. At ν=0\nu=0 both methods are identical. As ν\nu increases, the blob method produces increasingly non-Gaussian densities, while SBTM equilibrates correctly.
Figure 13: Weibel instability, dv=3d_{v}=3 : (v1,v2)(v_{1},v_{2}) -marginal density across collision frequencies. The blob method’s equilibration failure is more pronounced than in dv=2d_{v}=2 (Figure 12 ), indicating that SBTM scales better to higher dimensions.
 Runtime ( n=106n=10^{6} )Peak Memory (GB)
MethodTimeSpeedupn=106n=10^{6}n=3×106n=3\times 10^{6}
Blob (KDE)11h 27m–16.465.1
SBTM7h 17m1.57×1.57\times4.417.2
Table 2: Runtime and memory comparison for the two-stream instability test case ( M=100M=100 , Δt=0.05\Delta t=0.05 , dv=3d_{v}=3 , tfinal=50t_{\mathrm{final}}=50 ).
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