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Sparse model identification enables nonlinear dynamical system discovery fromdata. However, the control of false discoveries for sparse model identificationis challenging, especially in the low-data and high-noise limit. In this paper,we perform a theoretical study on ensemble sparse model discovery, which showsempirical success in terms of accuracy and robustness to noise. In particular,we analyse the bootstrapping-based sequential thresholding least-squaresestimator. We show that this bootstrapping-based ensembling technique canperform a provably correct variable selection procedure with an exponentialconvergence rate of the error rate. In addition, we show that the ensemblesparse model discovery method can perform computationally efficient uncertaintyestimation, compared to expensive Bayesian uncertainty quantification methodsvia MCMC. We demonstrate the convergence properties and connection touncertainty quantification in various numerical studies on synthetic sparselinear regression and sparse model discovery. The experiments on sparse linearregression support that the bootstrapping-based sequential thresholdingleast-squares method has better performance for sparse variable selectioncompared to LASSO, thresholding least-squares, and bootstrapping-based LASSO.In the sparse model discovery experiment, we show that the bootstrapping-basedsequential thresholding least-squares method can provide valid uncertaintyquantification, converging to a delta measure centered around the true valuewith increased sample sizes. Finally, we highlight the improved robustness tohyperparameter selection under shifting noise and sparsity levels of thebootstrapping-based sequential thresholding least-squares method compared toother sparse regression methods.

Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery