Sparse model identification enables nonlinear dynamical system discovery from data. However, the control of false discoveries for sparse model identification is challenging, especially in the low-data and high-noise limit. In this paper, we perform a theoretical study on ensemble sparse model discovery, which shows empirical success in terms of accuracy and robustness to noise. In particular, we analyse the bootstrapping-based sequential thresholding least-squares estimator. We show that this bootstrapping-based ensembling technique can perform a provably correct variable selection procedure with an exponential convergence rate of the error rate. In addition, we show that the ensemble sparse model discovery method can perform computationally efficient uncertainty estimation, compared to expensive Bayesian uncertainty quantification methods via MCMC. We demonstrate the convergence properties and connection to uncertainty quantification in various numerical studies on synthetic sparse linear regression and sparse model discovery. The experiments on sparse linear regression support that the bootstrapping-based sequential thresholding least-squares method has better performance for sparse variable selection compared to LASSO, thresholding least-squares, and bootstrapping-based LASSO. In the sparse model discovery experiment, we show that the bootstrapping-based sequential thresholding least-squares method can provide valid uncertainty quantification, converging to a delta measure centered around the true value with increased sample sizes. Finally, we highlight the improved robustness to hyperparameter selection under shifting noise and sparsity levels of the bootstrapping-based sequential thresholding least-squares method compared to other sparse regression methods.

翻译：稀疏模型识别能够从数据中发现非线性动力学系统。然而，控制稀疏模型识别中的假发现颇具挑战性，尤其是在低数据量和高噪声条件下。本文对集成稀疏模型发现进行了理论研究，该方法在准确性和对噪声的鲁棒性方面已展现出实证成功。具体而言，我们分析了基于自助法的序贯阈值最小二乘估计器，证明这种基于自助法的集成技术能够实现可证明正确的变量选择过程，且错误率呈指数收敛速度。此外，我们指出，与通过MCMC进行昂贵的贝叶斯不确定性量化方法相比，集成稀疏模型发现方法能以计算高效的方式进行不确定性估计。我们通过合成稀疏线性回归和稀疏模型发现的多种数值研究，展示了其收敛性质及与不确定性量化的关联。稀疏线性回归实验表明，与LASSO、序贯阈值最小二乘及基于自助法的LASSO相比，基于自助法的序贯阈值最小二乘方法在稀疏变量选择上具有更优性能。在稀疏模型发现实验中，我们证明基于自助法的序贯阈值最小二乘方法能够提供有效的不确定性量化，随着样本量增加，其收敛到以真值为中心的狄拉克测度。最后，我们强调，与其他稀疏回归方法相比，基于自助法的序贯阈值最小二乘方法在变化的噪声水平和稀疏度条件下对超参数选择具有更强的鲁棒性。