## 1 Introduction  
The liquid rocket engine, as the core component of a spacecraft, plays a crucial role in ensuring the safe launch and flight of a rocket. However, due to prolonged operation under harsh conditions such as high temperatures, high pressures, strong corrosion, and intense energy release, the engine is prone to malfunctions. If anomalies occur in the liquid rocket engine during ignition or flight, it often leads to rapid consequences such as explosions, resulting in significant financial losses and even jeopardizing human lives. Hence, it is imperative to conduct fault detection for liquid rocket engines. Engine fault detection essentially involves processing and analyzing multidimensional sensor time-series data, extracting features to obtain engine states. To detect the fault of LRE, statistical analysis methods such as red-line threshold algorithms [1] and autoregressive moving average (ARMA) algorithms [2] were initially applied due to their rapid and reliable computation. However, these methods often neglect the relationships between sensors, leading to false alarms due to the unstable threshold ranges and data of some sensors. With the advancement of deep learning techniques, an increasing number of deep neural models are employed in liquid rocket engine for fault detection, which aim to overcome the shortcomings of traditional machine learning methods such as artificial neural network (ANN) [3] and support vector machines (SVM) [4], which have inadequate feature extraction capabilities. [PERSON] et al. [3] integrated convolutional neural networks (CNN) and long short-term memory networks (LSTM) to identify transient faults during engine startup. [PERSON] et al. [5] proposed a memory-enhanced skip-connection autoencoder for unsupervised anomaly detection in rocket engine operations. [PERSON] et al. [6] employed a generative adversarial network (GAN) to achieve multi-source fusion for LRE anomaly detection. However, these approaches exhibit limited capacity in modeling long-term dependencies in multidimensional complex temporal data and are prone to overfitting issues due to sample imbalancesand small sample sizes. Moreover, the multi-source data fusion strategies are relatively simple and lack interpretability, hindering the representation of the relationship between sensors and model performance. Meanwhile, model-based fault detection methods are also investigated, utilizing mathematical analytical models of the engine combined with methods like Kalman filtering to achieve fault detection [7; 8]. The effectiveness of this approach depends on the accuracy of the model, yet analytical models established often diverge from actual engine behavior. Similar to model-based methods, state-space models describe the evolution of system states over time through state equations and observation equations. By integrating deep learning, state-space models can leverage their multi-level representation capabilities to capture features across different scales, thus enabling a more precise understanding and prediction of complex time series data. In 2022, [PERSON] et al. [9] employed state-space models for currency price forecasting. Similarly, in the same year, [PERSON] et al. [10] utilized state-space models for anomaly detection in time-series data. However, state-space models typically assume a linear relationship between system states and observations, which may pose limitations when dealing with the complex nonlinear relationships in liquid rocket engine data. Therefore, state-space models require further improvement to accommodate these complex scenarios. To tackle the aforementioned challenges, a novel approach named interpretable recurrent variational state-space model (IRVSSM) is proposed for fault diagnosis of liquid rocket engines, which leverages the recursive nonlinear state-space framework of deep neural networks. The nonlinear emission and transition matrices are designed to adeptly capture the intricate temporal dependencies and dynamic variations inherent among sequential signals.