## 1 Introduction  
In the ever-evolving landscape of semiconductor and solar photovoltaic technology, monocrystalline silicon stands as a material of paramount importance. Its production predominantly relies on the Czochralski process, a sophisticated and multi-staged method encompassing seeding, shoulder formation, body growth, and tailing. The basic principle is to use the molten silicon material to contact a small monocrystalline silicon seed crystal, and slowly pull the seed crystal at a specific temperature, so that the molten silicon solidifies on the surface of the seed crystal to form monocrystalline silicon. During the whole process, the temperature of the molten silicon is kept slightly higher than the melting point of silicon, and the uniformity and crystallization quality in the crystal growth process are promoted by rotating the seed crystal and/or furnace. The tail process reduces the crystal diameter to a certain extent by slowly increasing the temperature and crystal pulling speed, and finally leaves the silicon melt level. In the tail process, if the crystal diameter is not properly controlled, it will lead to crystal dislocation and crystal growth failure [1; 2; 3]. A schematic diagram of the Czochralski growth of monocrystalline silicon is shown in Figure 1.  
For crystal growth, it is necessary to ensure the integrity of crystal growth. Integrity means that in the tail of crystal growth, the tail diameter is continuously reduced until it is separated from the silicon melt. The integrity of the crystal tail can effectively prevent the reverse extension of crystal dislocations, thus damaging the quality of the wafer at the body stage. The different tail shapes are shown in Figure 2. Therefore, the tail diameter must be effectively controlled, and the premise of control is to measure the tail diameter in real time. Generally, we want the tail length to be as short as possible to save silicon. However, in the tail process, the crystal diameter obtained by image measurement [4] needs to ensure a longer tail length so that the crystal meniscus is visible. However, a longer tail length not only wastes silicon but also takes a large amount of time. Similarly, the method of using high-temperature infrared pyrometers to obtain the crystal diameter has the same disadvantages. Also, taking the crystal mass variation to estimate the crystal diameter of the tail growth still has many shortcomings. On the one hand, due to the detection accuracy and observation noise of the sensor, and on the other hand, due to the time-varying shape of the solid-liquid interface in the tail process [5], the estimation results are inaccurate. Therefore, we need to study the indirect measurement technology of the crystal diameter. The realm of soft measurement has witnessed significant advancements in recent years. In various studies, innovative approaches have been adopted, such as introducing dropout regularization into deep convolution networks for enhanced accuracy in health status identification [6], employing mutual information-based methods for effective biochemical oxygen demand prediction [7], and utilizing fuzzy least squares support vector machines optimized by adaptive genetic algorithms for torque measurement in AC dynamome  
Figure 1: A schematic diagram of the Czochralski growth of monocrystalline silicon. Figure 2: Tail shape of a crystal. ters [8]. In addition, in references [9; 10], the forecast models and wavelet neural network models have been applied to the field of soft measurement. However, these existing techniques, while groundbreaking, fall short in addressing the specific challenges posed by the Czochralski process, particularly in accurately measuring the tail diameter amid fluctuating process conditions. This highlights a significant gap in the existing methodologies, which need a more precise, non-intrusive, and efficient approach to measure the tail diameter.