Extreme radiation and high-energy-density conditions in laser–matter interactions demand precise numerical tools for modeling relativistic particle dynamics while strictly conserving key invariants such as energy and phase-space volume. Conventional particle pushers exhibit secular energy drift in ultra-intense fields where the normalized vector potential is large, leading to spurious heating artifacts that obscure genuine physical processes like laser-driven plasma wakefield acceleration and ion-cyclotron heating in traps. This work introduces a novel PINN–symplectic integrator hybrid, in which physics-informed neural networks trained on Hamiltonian structure are combined with geometric symplectic schemes to enforce near-exact long-term energy conservation. The method solves the relativistic equations of motion in laboratory coordinates and incorporates arbitrary time-dependent electromagnetic fields through neural surrogates for field interpolation, achieving high accuracy without resorting to costly adaptive time-stepping. Validation benchmarks include single-particle gyromotion in crossed electric and magnetic fields mimicking ion traps, where the energy error remains below one part in ten thousand after one million gyroperiods, as well as laser wakefield acceleration in underdense plasma at high intensities, where dephasing and energy gain are reproduced with high fidelity. Additional high-energy-density test cases, such as fast-ignition-relevant electron heating, demonstrate that the scheme clearly distinguishes field-induced energization from numerical diffusion. Multi-particle extensions implemented via ensemble methods scale linearly on GPUs and enable simulations with millions of particles, making the approach suitable for realistic modeling of extreme matter and radiation environments and for interpreting experiments at present and next-generation petawatt-class laser facilities.

symplectic integrators; physics-informed neural networks; relativistic particle dynamics; laser–plasma interactions; energy conservation; high-energy-density physics