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

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.

Score-Based Deterministic Density Sampling

We propose and analyze a deterministic sampling framework using Score-Based Transport Modeling (SBTM) for sampling an unnormalized target density $\pi$. While diffusion generative modeling relies on pre-training the score function $\nabla \log f_t$ using samples from $\pi$, SBTM addresses the more general and challenging setting where only $\nabla \log\pi$ is known. SBTM approximates the Wasserstein gradient flow on KL$(f_t\|\pi)$ by learning the time-varying score $\nabla \log f_t$ on the fly using score matching. The learned score gives immediate access to relative Fisher information, providing a built-in convergence diagnostic. The deterministic trajectories are smooth, interpretable, and free of Brownian-motion noise, while having the same distribution as ULA. We prove that SBTM dissipates relative entropy at the same rate as the exact gradient flow, provided sufficient training. We further extend our framework to annealed dynamics, to handle non log-concave targets. Numerical experiments validate our theoretical findings: SBTM converges at the optimal rate, has smooth trajectories, and is easily integrated with annealed dynamics. We compare to the baselines of ULA and annealed ULA.

Target Tracking using Robust Sensor Motion Control

We consider the problem of tracking moving targets using mobile wireless sensors (of possibly different types). This is a joint estimation and control problem in which a tracking system must take into account both target and sensor dynamics. We make minimal assumptions about the target dynamics, namely only that their accelerations are bounded. We develop a control law that determines the sensor motion control signals so as to maximize target resolvability as the target dynamics evolve. The method is given a tractable formulation that is amenable to an efficient search method and is evaluated in a series of experiments involving both round-trip time based ranging and Doppler frequency shift measurements

Transport based particle methods for the Fokker-Planck-Landau equation

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.