Data-Driven Reconstruction and Characterization of Stochastic Dynamics via Dynamical Mode Decomposition

Noise fundamentally limits the performance and predictive capabilities of classical and quantum dynamical systems by degrading stability and obscuring intrinsic dynamical characteristics. Characterizing such noise accurately is essential for enhancing measurement precision, understanding environmental interactions, and designing effective control strategies across diverse scientific and engineering domains. However, extracting spectral features and associated characteristic decay or coherence times from limited and noisy datasets remains challenging. Here, we introduce a general, data-driven framework based on Dynamical Mode Decomposition (DMD) to analyze system dynamics under stochastic noise. We reinterpret DMD modes as statistical weights over ensembles of stochastic trajectories, using a nonlinear transformation to construct noise power spectral densities (PSDs). This enables the identification of dominant frequency contributions in both broadband (white) and correlated ($1/f$) noise environments, as well as direct extraction of intrinsic characteristic decay times from DMD eigenvalues. To overcome instability in standard DMD-based extrapolation, we develop a constrained reconstruction method using extracted decay times as physical bounds and the learned PSD as weights. We demonstrate the effectiveness of this approach through simulations of quantum system dynamics subjected to decoherence from noise, validating its robustness and predictive capabilities. This methodology provides a broadly applicable tool for diagnostic, predictive, noise mitigation analyses, and control in complex stochastic systems.