Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations

Abstract Out-of-equilibrium quantum many-body systems—such as multi-electron atoms and molecules driven by strong laser fields, quenched ultracold gases, and ultrafast-excited solids—exhibit rapid correlation buildup that underlies many emerging phenomena. Exact wave-function methods to describe these effects scale exponentially with particle number; simpler mean-field approaches neglect essential particle correlations. The time-dependent two-particle reduced density matrix (TD2RDM) formalism offers a middle ground by propagating the two-particle density matrix and closing the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy with a reconstruction of the three-particle cumulant, which carries information about three-particle correlations. But the validity and existence of time-local reconstruction functionals ignoring memory effects remain unclear across different dynamical regimes. We show that a neural ordinary differential equation (ODE) model trained on exact two-particle reduced density matrix (2RDM) data (no dimensionality reduction) can reproduce its full dynamics without any explicit three-particle information—but only in parameter regions where the Pearson correlation between the two- and three-particle cumulants is large. In contrast, in the anti-correlated or uncorrelated regime, the neural ODE fails, indicating that no simple time-local functional of the instantaneous two-particle cumulant can capture the evolution. The magnitude of the time-averaged three-particle-correlation buildup appears to be the primary predictor of successful extrapolation: for a moderate correlation buildup, both neural ODE predictions and existing TD2RDM reconstructions are accurate, whereas stronger values lead to systematic breakdowns. These findings pinpoint the need for memory-dependent kernels in the three-particle cumulant reconstruction for the latter regime. Our results place the neural ODE as a model-agnostic diagnostic tool that maps the regime of applicability of cumulant expansion methods and guides the development of non-local closure schemes. More broadly, the ability to learn high-dimensional reduced-density-matrix dynamics from limited data opens a pathway to fast, data-driven simulation of correlated quantum matter, complementing traditional numerical and analytical techniques.