Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black-box models is that they underperform under blind conditions since no physical knowledge is incorporated. Physics-based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics-based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics-based model with respect to the input and parameters and shares similarity with the exact Autocovariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes originating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated.

翻译：机器学习已广泛应用于物理系统的建模与预测。这些技术在观测数据集的插值任务中表现出强大的表达能力和良好的泛化性。然而，黑箱模型的缺陷在于未融入物理知识，导致其在无先验条件下的性能不佳。基于物理的机器学习旨在通过保留机器学习技术的数学灵活性同时引入物理知识来解决这一问题。为此，本文提出将基于力学的模型嵌入高斯过程的均值函数，并通过核函数表征潜在偏差。我们推广了一类特定核函数，该函数与物理模型关于输入和参数的梯度存在关联，并与线性动力系统的精确自协方差函数具有相似性。此类核函数的谱特性能够识别因物理模型设定错误而产生的占优周期过程。然而，在处理长序列数据时，核函数的平稳性成为严峻障碍，本文通过分层贝叶斯技术加以解决。该实现方法还有助于降低计算成本，缓解高斯过程在处理序列数据时的可扩展性问题。通过数值与实验案例，展示了所提方法在结构动力学逆问题中的潜在应用。