Data-driven higher-order moment inverse modeling for jump–diffusion dynamics

In complex engineering and aerospace systems, experimental repeatability, operating cost, and safety constraints often limit data acquisition to a single long-time realization, making closed-form stochastic equation discovery from finite-sampling-rate observations a challenging inverse problem. This difficulty becomes more pronounced when continuous Brownian fluctuations and intermittent jump events coexist, as finite-lag conditional statistics mix drift, diffusion, and jump-induced contributions while nonuniform state-space occupancy increases the uncertainty of higher-order estimates. To address these issues, this paper proposes the Cumulant-based Adaptive Stochastic-System Identification (CASSI) framework for reconstructing multidimensional stochastic jump–diffusion systems from higher-order conditional statistics. CASSI employs an adaptive Nadaraya–Watson (NW) estimator to estimate finite-lag conditional increment statistics under uneven state-space sampling and converts the estimated raw moment fields into finite-lag plug-in cumulant-rate estimates that represent the corresponding connected higher-order contributions. These cumulant-based targets support the identification of jump intensity and jump-amplitude characteristics and enable the separation of drift, continuous diffusion, and jump-induced contributions. The resulting coefficient fields are subsequently converted into compact analytical expressions by symbolic regression without prescribing a fixed candidate-function library. Validations on one- and two-dimensional systems with smooth and nonsmooth drift terms under four representative jump–diffusion scenarios demonstrate accurate closed-form reconstruction from a single trajectory with a known sampling interval. Comparisons with existing benchmark methods show improved coefficient-field and jump-parameter identification accuracy under finite sampling intervals and pronounced non-Gaussian jump effects. The reconstructed equations reproduce the principal dynamical response characteristics and stationary probability distributions of the reference systems, indicating that CASSI provides a statistically grounded and interpretable route for closed-form identification of multidimensional non-Gaussian stochastic dynamics.