An Adaptive Three-Stage Recovery Framework for Sparse Identification of Nonlinear Dynamical Systems

Sparse identification of nonlinear dynamical systems provides a powerful framework for signal processing when governing equations remain unknown. Feature-selection-based methods must balance parsimony with accurate representation. However, existing methods typically perform selection in a single pass and permanently exclude candidates without reconsideration, which is problematic when measurement noise or weak signal strength causes true governing terms to be incorrectly pruned. To address this issue, we introduce an adaptive three-stage framework with an integrated recovery mechanism. First, an adaptive thresholding stage reduces the candidate feature library while preserving potential governing terms. Second, a forward-backward stepwise selection guided by the Bayesian information criterion provides a theoretically grounded model selection mechanism without requiring specification of regularization parameters. Third, a residual-based post-audit examines residual correlations to identify potentially missed features, and bootstrap resampling removes terms with unreliable coefficient estimates. Extensive experiments on synthetic systems, partial differential equations and real-world applications are conducted to demonstrate the effectiveness of the proposed method.