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 the environment 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 an ensemble of stochastic trajectories and, via a nonlinear mapping, construct a PSD-like spectral fingerprint of the noise. 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 noise as spectral weights. We demonstrate the effectiveness of this approach through simulations of quantum system dynamics subject to decoherence from noise, demonstrating its robustness and predictive capabilities and comparing it with standard methods. This methodology provides a trajectory-level framework for diagnostic and predictive analysis of stochastic processes from limited, noisy time-series data.