## 1 Introduction  
It is essential to keep a system safe to minimize downtime and avoid catastrophic failures. To achieve this, effective maintenance strategies have been developed, and recent attention has shifted toward predictive maintenance (PdM) that relies on the predicted health condition of a system [1, 2, 3]. Central to PdM is prognostics, which forecasts a system's future degradation behavior and remaining useful life (RUL) based on its health monitoring data. Recent advancements in big data analytics, machine learning, and artificial intelligence have significantly influenced the development of prognostic techniques. Prognostic methods can generally be categorized into physics-based [4] and data-driven [5] approaches, depending on the type of information used. In physics-based approaches, the key element is a physical model that describes degradation behavior over time under various loading conditions. For example, the Paris model [6], which describes crack growth behavior, is a representative physical degradation model. In such approaches, the primary concern is identifying model parameters based on the available data. Common algorithms for estimating these parameters include particle filters [7] and the Bayesian Markov chain Monte Carlo method [8]. [PERSON] and [PERSON] [9] and [PERSON] et al. [10] employed a particle filter-based framework and a Bayesian method, respectively, to predict crack growth and the RUL. Although physics-based approaches can provide accurate predictions of degradation behavior and RUL with relatively small datasets, recent research on this topic is limited due to the scarcity of physical degradation models. Consequently, there has been growing interest in data-driven approaches. Data-driven approaches use monitoring data and mathematical models selected based on the characteristics of the data. Since the use of mathematical models is flexible, various algorithms are available, including nonlinear least squares [11], Gaussian processes [12], and artificial neural networks (ANNs) [13]. Among these, ANNs and their variations have been extensively employed for data-driven prognostics. For instance, [PERSON] et al. [14] proposed a data-driven method based on long short-term memory (LSTM) networks, a type of recurrent NN, and validated it on the C-MAPSS dataset for aerospace engines. Using the same dataset, [PERSON] et al. [15] proposed a 1D convolutional NN-LSTM model with difference-based features, demonstrating superior performance on the C-MAPSS FD001 dataset. Additionally, [PERSON] et al. [16] developed a gated recurrent unit-based method for hydrogen fuel cell RUL prediction, outperforming the back propagation NN and LSTM on the FCLAB datasets. Despite the advancements in data-driven approaches, the limited availability and quality of degradation data have motivated the development of hybrid approaches. One such approach is physics-informed neural networks (PINNs) [17], which embed physical laws into the training of ANNs as a form of regularization to enhance predictive accuracy. [PERSON] et al. [18] employed a PINN that incorporates attributes affecting battery degradation, based on an empirical degradation equation, for the accurate and stable estimation of battery state of health. [PERSON] et al. [19] implemented physics-informed machine learning to demonstrate improvements in the precision of RUL prediction, achieving faster learning convergence and better generalizability. Although it is desirable to develop sophisticated algorithms to improve accuracy with less information, existing prognostic algorithms remain difficult to understand and implement, and their accuracy levels are often unsatisfactory given their complexity. Understanding these algorithms requires knowledge of several mathematical and physics theories, such as Bayesian statistics, probability theory, numerical sampling methods, optimization techniques, and partial differential equations. Moreover, as the complexity of a prediction model increases, so do the computational time and cost.