SINDy Enhanced for Noisy Data with Koopman Upsampling

Identifying the underlying governing equations of dynamic systems from observed data is a critical task in scientific machine learning, often performed using methods like Sparse Identification of Nonlinear Dynamics (SINDy) and PDE-FIND. A major challenge arises when data is sparse and noisy, as this can lead to unreliable derivative estimates, which are crucial for these identification techniques. This paper addresses this issue by integrating Koopman-based upsampling methods, including Dynamic Mode Decomposition (DMD) and Extended DMD (EDMD), as a preprocessing step. These techniques learn finite-dimensional approximations of Koopman evolution, allowing for the interpolation and denoising of data snapshots within the observed time window. This dynamics-aware preprocessing significantly reduces derivative estimation errors, leading to more accurate recovery of ordinary and partial differential equations. Experimental evaluations on various ODE and PDE systems demonstrate that Koopman-based upsampling, particularly polynomial EDMD for ODEs, consistently improves performance over non-dynamical interpolation methods, especially in sparse and noisy regimes.