Ensemble adaptive libraries and inner-product sparse regression for robust nonlinear dynamics discovery

Identifying nonlinear dynamics from noisy data is a challenging yet fundamental task. Sparse Identification of Nonlinear Dynamical Systems (SINDy) and its variant, Weak SINDy (WSINDy), are widely used approaches. SINDy enables efficient sparse modeling but is hindered by sensitivity to noise, dependence on explicit time-derivative estimation, and reliance on fixed feature libraries. WSINDy improves noise resilience by using integral formulations but introduces new challenges related to test function selection and computational overhead. To address these issues, we introduce EALMM-ipSINDy, a framework that integrates linear multistep methods, ensemble adaptive libraries, and inner-product-driven sparse regression. Our approach avoids explicit derivative estimation by directly approximating dynamics with linear multistep methods, while adaptive libraries are constructed through incremental feature adaptation and ensemble cross-validation to ensure accuracy with minimal redundancy. An inner-product-driven sparse regression algorithm, equipped with cross-validation, then extracts the most informative features. In addition, we design a theoretically grounded adaptive moving average filter that provides superior noise reduction compared to standard denoising methods. Numerical experiments demonstrate that our framework significantly improves accuracy over SINDy, moderately outperforms WSINDy, and achieves lower computational complexity, thereby offering a robust and effective solution for recovering nonlinear dynamics from noisy data.