This paper addresses the issue of overfitting while calibrating unknown parameters of over-parameterized physics-based models with noisy and incomplete observations. A semi-analytical Bayesian framework of nonlinear sparse Bayesian learning (NSBL) is proposed to identify sparsity among model parameters during Bayesian inversion. NSBL offers significant advantages over machine learning algorithm of sparse Bayesian learning (SBL) for physics-based models, such as 1) the likelihood function or the posterior parameter distribution is not required to be Gaussian, and 2) prior parameter knowledge is incorporated into sparse learning (i.e. not all parameters are treated as questionable). NSBL employs the concept of automatic relevance determination (ARD) to facilitate sparsity among questionable parameters through parameterized prior distributions. The analytical tractability of NSBL is enabled by employing Gaussian ARD priors and by building a Gaussian mixture-model approximation of the posterior parameter distribution that excludes the contribution of ARD priors. Subsequently, type-II maximum likelihood is executed using Newton's method whereby the evidence and its gradient and Hessian information are computed in a semi-analytical fashion. We show numerically and analytically that SBL is a special case of NSBL for linear regression models. Subsequently, a linear regression example involving multimodality in both parameter posterior pdf and model evidence is considered to demonstrate the performance of NSBL in cases where SBL is inapplicable. Next, NSBL is applied to identify sparsity among the damping coefficients of a mass-spring-damper model of a shear building frame. These numerical studies demonstrate the robustness and efficiency of NSBL in alleviating overfitting during Bayesian inversion of nonlinear physics-based models.

本文解决了在过度嘈杂和不完整观测值校准过参数化基于物理学的模型的未知参数时的过度拟合问题。提出了一种非线性稀疏贝叶斯学习（NSBL）的半解析贝叶斯框架，以识别贝叶斯反演期间模型参数之间的稀疏性。对于基于物理学的模型，NSBL与稀疏贝叶斯学习（SBL）的机器学习算法相比具有显着优势，例如1）似然函数或后验参数分布不需要是高斯的； 2）先验参数知识已纳入稀疏学习（即，并非所有参数都被视为有问题）。NSBL采用自动相关性确定（ARD）的概念，通过参数化的先验分布来促进可疑参数之间的稀疏性。通过采用高斯ARD先验并通过建立后参数分布的高斯混合模型近似来排除NSARD的贡献，可以实现NSBL的分析可处理性。随后，使用牛顿法执行II类最大似然，从而以半分析方式计算出证据及其梯度和黑森州信息。我们通过数值和分析表明，SBL是线性回归模型中NSBL的特例。随后，考虑在参数后pdf和模型证据中均涉及多峰的线性回归示例，以证明NSBL在SBL不适用的情况下的性能。接下来，应用NSBL来识别剪切建筑框架的质量-弹簧-阻尼器模型的阻尼系数中的稀疏性。