In the case of physical asset monitoring, there are two main branches of modeling: physics-based modeling, also known as physical-law modeling, and data-driven modeling. Physics-based modeling aims to describe phenomena by formulating mathematical models that integrate interdisciplinary knowledge to generate models that replicate observed behavior. Models are commonly presented in differential equations whose complexity increases as more factors become involved. Several terms and parameters must be defined to fully describe the system phenomena. Initial and boundary conditions must be identified to obtain physics equation solutions, and the computational cost associated with this operation can be very time-consuming for complex phenomena.

In SHM systems, these models are implemented to assess the condition of an asset under operating conditions to monitor changes that may indicate the presence of damage and shorten the remaining useful life of the asset. Finite element modeling (FEM) software implements well-known analyses, such as modal analysis, and allows the simulation of different structures and initial and boundary conditions straightforwardly. FEM software includes physical law models integrated into software libraries to perform damage analysis on a virtualized model of structures efficiently.

The complexity of modeling building structures under seismic excitation is caused by the intervening factors that can modify the behavior of these structures and by the difficulty of correctly defining their physical properties. Several works focus on estimating model parameters and uncertainties to improve the model of the structure. For example, Xu et al. [55] estimated the parameters of the structures based on linear and nonlinear regression analyses. These structures’ linear and nonlinear parameters, such as elastic stiffness and yield displacement, are obtained for a three-story structure. Gomes et al. [56] addressed an inverse identification problem using numerical models and a genetic algorithm.

Model parameters obtained from experimental and recorded data can increase the model’s accuracy compared to the actual measured results. However, environmental and operating conditions do not remain constant throughout the life of the structure. In addition, physical law models have drawbacks that limit their applications in some SHM systems. The time to solve the equations in a real-time SHM system impacts the response time to ensure safety and the reduction of economic losses in response systems for seismic protocols and evacuation procedures. In order to reduce the time costs of performing calculations, optimization algorithms applied to the problem of SHM are encouraged and proposed in the literature [57].

Table 6 lists physics-based proposals for SHM systems in buildings under vibration excitation for multi-story structures. In this table, two types of proposal contributions are shown: the identification of system parameters and damage. According to Farrar’s classification, the level of damage identification for the practical proposals is also presented.

Machine-learning techniques are a subset of the field of artificial intelligence. Due to their statistical nature, their vision is to address problems of interest in pattern recognition identification and classification tasks. SHM from the ML point of view is a classification problem in which at least two states are compared in SHM systems employing ML techniques: damaged and undamaged states.

A machine-learning model extracts information in the form of features from a given data set and classifies those data features. These data-driven models require large amounts of information to train the model and avoid the overfitting problem. The generalization problem depends on the amount of available data and significant diversity of this training data to avoid overfitting and ensure a reasonable level of generalization. In an idealized situation, the data set should include the samples of the possible range of excitation that can be applied to the structure. In addition, data quality improvement using signal processing techniques, such as normalization and noise filtering, is desirable for the generation of the data set. ML algorithms are applied in the damage identification process and in analyzing the anomalous data obtained from the sensors [8,65], thereby improving data quality. In addition, signal processing techniques, such as WT and HHT also improve data quality and are applied in SHM systems [9,66].

ML techniques can be divided into supervised learning (for regression and classification tasks), unsupervised learning (anomaly detection and clustering) and reinforcement learning. The most popular ML techniques implemented in the construction of SHM solutions are support vector machines (SVMs) and convolutional neural networks (CNNs) [10,67]. In the case of neural network techniques, damage-sensitive feature selection plays a crucial role in the performance of the SHM system. SVM optimally classifies features in linear and nonlinear problems. CNN, a subset of neural network (NN) methods, include convolution operations in the hidden layer of neural networks to classify data, usually in image format. Other techniques, such as PCA, improve the features of the training data set by making them uncorrelated.

The selection of a ML technique is guided by the limitations of each technique and the requirements of SHM in terms of damage identification level and operating conditions. Identifying the modal systems of building structures can also be performed using deep neuronal networks (DNNs) [11,68]. Table 7 summarizes the ML-based proposals for SHM systems in buildings under vibration excitation for multi-story structures.