This study presents a Bayesian maximum \textit{a~posteriori} (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized zeroth-order Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint maximum \textit{a~posteriori} (JMAP) and variational Bayesian approximation (VBA), are compared to the popular SINDy algorithm for thresholded least-squares regression, by application to several dynamical systems with added noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or ``Gaussian norm'' $||\vy-\hat{\vy}||^2_{M^{-1}} = (\vy-\hat{\vy})^\top {M^{-1}} (\vy-\hat{\vy})$, where $\vy$ is a vector variable, $\hat{\vy}$ is its estimator and $M$ is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection.

翻译：本研究提出一种基于贝叶斯最大后验(MAP)框架的方法,用于从时间序列数据中识别动力系统。该方法被证明等价于广义零阶Tikhonov正则化,通过负对数似然与先验分布分别对残差项和正则化项的选择提供了理论依据。除模型系数估计外,贝叶斯解释还可访问贝叶斯推断的全部工具,包括模型排序、模型不确定性量化及未知(干扰)超参数估计。通过将联合最大后验(JMAP)和变分贝叶斯近似(VBA)两种贝叶斯算法应用于多个含噪动力系统,将其与流行的基于阈值最小二乘回归的SINDy算法进行对比。对于多变量高斯似然与先验分布,贝叶斯公式给出高斯后验与证据分布,其分子项可用马氏距离或"高斯范数"$||\vy-\hat{\vy}||^2_{M^{-1}} = (\vy-\hat{\vy})^\top {M^{-1}} (\vy-\hat{\vy})$表示,其中$\vy$为向量变量,$\hat{\vy}$为其估计量,$M$为协方差矩阵。研究表明后验高斯范数为定量模型选择提供了稳健度量指标。