Symmetry-Preserving Physics-Informed Neural Network Framework for Relativistic Charged-Particle Dynamics in 3+1 Dimensions

Standard pushers for the relativistic equations of motion of a charged particle in an electromagnetic field—Boris, Vay, Higuera–Cary—do not, in general, preserve the full symplectic structure of the underlying Hamiltonian system, while high-order non-symplectic schemes such as Runge–Kutta accumulate secular error over long times. We propose a symmetry-preserving physics-informed neural network framework (SP-PINN) for the 3+1-dimensional relativistic dynamics of a charged particle in a prescribed field, including a focused Gaussian laser pulse. The method is two-stage: an unsupervised physics-informed neural network learns a surrogate relativistic Hamiltonian from the covariant equations of motion using a Lorentz-invariant loss that enforces the mass-shell constraint H=mc²γ; the surrogate is then advanced with an explicit symplectic map built on Tao’s extended phase space, valid for the non-separable relativistic Hamiltonian. We benchmark against the Boris pusher and Runge–Kutta on three test problems. The magnetic-field test illustrates the contrast between bounded and secular error growth: Runge–Kutta drifts secularly, the Boris pusher conserves the invariants to machine precision as a volume-preserving gyro-integrator, and the symplectic map keeps the error bounded for all time; on a non-integrable magnetic trap, where no exact volume-preserving rotation exists, the symplectic map alone keeps the energy error bounded. The learned surrogate is the current accuracy bottleneck; for the demanding laser case a vector-potential light-cone reformulation reduces its error to (3.0±0.1)×10⁻⁴ (three seeds) and yields learned trajectories that remain phase-coherent over essentially the whole interaction. The framework targets laser–plasma acceleration, synchrotron-radiation modeling, and particle tracking.