Symplectic Symmetry Preservation in Relativistic Particle Dynamics via Physics-Informed Neural Network Integrators

The long-term numerical integration of relativistic charged particle dynamics in intense laser-plasma fields poses a fundamental challenge: conventional explicit integrators break the symplectic symmetry of the underlying Hamiltonian structure, leading to systematic energy drift over extended simulation times. This symmetry violation reflects a failure to preserve the phase-space geometry dictated by Liouville's theorem and the canonical Poisson bracket structure of relativistic mechanics.

We present a physics-informed neural network (PINN) framework explicitly designed to enforce symplectic symmetry as a hard architectural constraint. The network is trained to approximate the flow map of the relativistic Hamiltonian system, with the symplectic condition incorporated directly into the loss functional alongside the equations of motion residuals and initial condition penalties. This approach preserves the time-reversal and area-preserving symmetries of Hamiltonian dynamics by construction, in contrast to standard PINN formulations where physical constraints are imposed only softly. The method is validated for relativistic electrons in the fields of plane monochromatic waves and ultrashort pulsed laser beams, using rapidity-spectral Lorentz coordinates as the dynamical variables. Energy conservation errors are compared across classical integrators, standard symplectic methods, and the proposed PINN-symplectic scheme over timescales spanning many oscillation periods. The PINN-symplectic integrator demonstrates bounded energy error growth consistent with near-conservation of the modified Hamiltonian, outperforming both standard PINN and explicit Runge–Kutta approaches. These results demonstrate that embedding continuous symmetry principles into neural network architectures yields physically consistent, long-time-stable simulations of relativistic systems — a finding of direct relevance to laser-driven particle acceleration, spin-polarized beam modeling, and relativistic plasma diagnostics at next-generation facilities.