## 1 Introduction  
Due to the generation mechanism and harmful effects of nitrogen oxides (NOx) in the exhaust emissions of low-speed marine diesel engines, the International Maritime Organization and national regulations have strict NOx emission limits for marine engines. Low-speed marine engines are verified by emission tests to meet these emission regulations before being installed on board ships. However, as marine low-speed engines operate for long periods, environmental changes and the degradation of certain components such as fuel injectors and solenoid valves affect their emission performance. In order to ensure that marine diesel engines can operate efficiently with low emissions under different operating conditions, it is necessary to be able to obtain accurate parameter information during diesel engine operation; as a result, more and more sensors are being installed on marine diesel engines. Among the many sensors, the NOx sensor is a necessary tool for the detection of NOx emissions, which provides NOx data to SCR systems to ensure that all NOx emissions in the exhaust gas are reduced by urea as much as possible. However, in the process of practical application, the NOx sensor needs to be exposed to the engine exhaust gas for a long time, and impurities such as sulfides and particulate matter in the exhaust gas cause the poisoning of sensitive components and reduce the service life. Related papers show that the continuous measurement of ship engine exhaust gas will reduce the service life of NOx sensors to 85 h [1]. Due to the reduced lifetime of NOx sensors, the cost of use and the uncertainty in the application process are increased. In this situation, soft sensors or virtual sensors based on soft measurement techniques are being applied and developed. The basic idea of soft measurement technology to solve the lifetime problem in NOx sensor applications is to predict NOx emissions from diesel engines by modeling the correlation between NOx emissions and other directly measured or easily measurable variables without using NOx sensors directly. According to the principles of modeling NOx simulations, the models can be divided into physical mechanism models and data-driven models [2]. The mechanism-based NOx simulation model is still mainly based on the widely accepted Zeldovich mechanism as the NOx generation model. Based on constructing relevant variables, it can be divided into two categories: (1) Mathematical equations based on the working mechanism of the engine to form a more suitable NOx prediction through the appropriate simplification or additional consideration of the mixing and combustion conditions in the process of NOx formation [3; 4; 5; 6; 7; 8; 9; 10]; and (2) use some physical signals (such as in-cylinder pressure, torque, etc.) during the working process of the engine to obtain intermediate variables through signal analysis and processing, and then combine the Zeldovich mechanism to analyze and deduce the generation of NOx to construct NOx predictive models [11; 12; 13; 14; 15]. The method of the physical mechanism model needs to be familiar with the formation mechanism of NOx under the characteristics of different models and fuel characteristics and conduct strict physical formula derivation, which can describe the in-cylinder combustion and the formation process of NOx more completely. The solution process often requires iteratively solving mathematical and physical equations, and the results obtained have strong interpretability, but the solution process is complicated. Using empirical equations to simplify the calculation of parameters such as NOx can reduce the computational complexity, but cannot guarantee sufficient accuracy. The data-driven model uses artificial intelligence algorithms to build a corresponding calculation model to train and learn between input and output, and finally obtain a relatively accurate calculation model. It is considered to be an effective way of solving the prediction of difficult-to-measure physical quantities.