NEURAL FIELD-BASED CHARACTERISTIC RECONSTRUCTION FOR VLASOV–POISSON DYNAMICS

The particle distribution in Vlasov–Poisson dynamics develops increasingly fine structures over time, making the full distribution difficult for a neural network to represent directly, whereas the electric field driving particle trajectories depends only on position and time and usually remains comparatively smooth. Motivated by this difference, we introduce Neural Field-Based Characteristic Reconstruction (NFCR), in which the neural network represents only the electric field rather than directly learning the full distribution. A time network outputs a finite set of spatial Fourier coefficients over the training interval, and differentiable backward characteristic integration reconstructs the evolving distribution from the prescribed initial condition. The density and current obtained from this distribution are related to the network output through the Poisson and Amp` ere equations, keeping the electric field consistent with the reconstructed distribution; conditions on the initial field and its time derivative constrain the dynamics near the beginning of the interval. The neural network therefore needs to learn only a finite number of electric-field coefficients, while fine distribution scales are generated naturally by transport, without an evolving phase-space grid or reference labels. Backward trajectories are integrated with a symmetric drift–acceleration–drift scheme. This scheme locally preserves phase-space area during backward tracing and ensures that a distribution reconstructed from a nonnegative initial condition remains nonnegative. Experiments on Gaussian transport and Landau damping show that NFCR reconstructs the distribution with lower errors than neural methods that directly learn the distribution or the full particle-trajectory map, while accurately capturing the electric-field evolution and maintaining small mass drift.