A non-perturbative framework for ion stopping in partially ionized warm dense matter with a physics-informed neural-network solver

Abstract Ion energy deposition in dense plasmas is of central importance to ion-beam-driven inertial confinement fusion and high-energy-density physics. In the Bragg-peak regime, the stopping power governs the spatial distribution of deposited energy and therefore strongly affects the formation of warm dense states and the efficiency of ion fast ignition. In this regime, the projectile velocity becomes comparable to the characteristic velocity of target electrons, and the beam–plasma interaction may enter a nonlinear, non-perturbative scattering regime in which Born-based perturbative treatments are no longer adequate. For partially ionized warm dense matter, additional complexity arises from charge-state distributions, bound-electron response and dynamic screening. Here, we develop a physically constrained, self-consistent non-perturbative framework for ion stopping in warm dense matter. The approach combines formal quantum-scattering relations, a charge-state-resolved collisional–radiative model and multiscale effective interaction potentials. In the strong-scattering regime, non-perturbative phase shifts are obtained from the Calogero equation using a variable-phase physics-informed neural-network solver, while the Born limit is recovered naturally in the weak-coupling asymptotic regime. This provides a unified description in which deviations from perturbative stopping can be quantified and the onset of perturbative breakdown can be identified. The framework is validated against time-resolved proton energy-loss measurements in a hydrogen discharge plasma and against FCDF benchmarks in the classical and strongly degenerate limits. For partially ionized warm dense carbon, the calculated stopping power remains close to TD-KS-DFT and TD-OF-DFT reference data, while conventional phenomenological and average-atom models tend to overestimate the Bragg-peak region. These results indicate that higher-order phase-shift corrections are essential in the strongly coupled regime and that the present framework provides a consistent non-perturbative description of ion energy loss in partially ionized warm dense plasmas.