Sparse Identification of Nonlinear Dynamics

Data-driven discovery of governing equations from time-series data provides a powerful framework for understanding complex biological systems. Library-based approaches that use sparse regression over candidate functions have shown considerable promise, but they face a critical challenge when candidate functions become strongly correlated: numerical ill-conditioning. Poor or restricted sampling, together with particular choices of candidate libraries, can produce strong multicollinearity and numerical instability. In such cases, measurement noise may lead to widely different recovered models, obscuring the true underlying dynamics and hindering accurate system identification. Although sparse regularization promotes parsimonious solutions and can partially mitigate conditioning issues, strong correlations may persist, regularization may bias the recovered models, and the regression problem may remain highly sensitive to small perturbations in the data. We present a systematic analysis of how ill-conditioning affects sparse identification of biological dynamics using benchmark models from systems biology. We show that combinations involving as few as two or three terms can already exhibit strong multicollinearity and extremely large condition numbers. We further show that orthogonal polynomial bases do not consistently resolve ill-conditioning and can perform worse than monomial libraries when the data distribution deviates from the weight function associated with the orthogonal basis. Finally, we demonstrate that when data are sampled from distributions aligned with the appropriate weight functions corresponding to the orthogonal basis, numerical conditioning improves, and orthogonal polynomial bases can yield improved model recovery accuracy across two baseline models.

The stability and control of Unmanned Aerial Vehicles (UAVs) in a turbulent environment is a matter of great concern. Devising a robust control algorithm to reject disturbances is challenging due to the highly nonlinear nature of wind dynamics, and modeling the dynamics using analytical techniques is not straightforward. While traditional techniques using disturbance observers and classical adaptive control have shown some progress, they are mostly limited to relatively non-complex environments. On the other hand, learning based approaches are increasingly being used for modeling of residual forces and disturbance rejection; however, their generalization and interpretability is a factor of concern. To this end, we propose a novel integration of data-driven system identification using Sparse Identification of Non-Linear Dynamics (SINDy) with a Recursive Least Square (RLS) adaptive control to adapt and reject wind disturbances in a turbulent environment. We tested and validated our approach on Gazebo harmonic environment and on real flights with wind speeds of up to 2 m/s from four directions, creating a highly dynamic and turbulent environment. Adaptive SINDy outperformed the baseline PID and INDI controllers on several trajectory tracking error metrics without crashing. A root mean square error (RMSE) of up to 12.2 cm and 17.6 cm, and a mean absolute error (MAE) of 13.7 cm and 10.5 cm were achieved on circular and lemniscate trajectories, respectively. The validation was performed on a very lightweight Crazyflie drone under a highly dynamic environment for complex trajectory tracking.

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.