## 2 Related Works
### Combination of Model-Based Methods and Data-Driven Methods  
Neural network has a powerful nonlinear modeling ability and can accomplish tasks that are difficult for traditional methods. Data-based neural network modeling approaches rely heavily on data. When dealing with some practical problems, the existing prior knowledge can often be used to improve the effectiveness of the model. Merging model-based and data-driven methods is an effective way to use prior knowledge. [PERSON] et al. [30] proposed the Hamiltonian Neural Networks with knowledge of exact conservation laws, which can accelerate the training process of the model and have better generalization than a regular neural network. [PERSON] et al. [31] proposed the APHYNITY framework for augmenting incomplete physical dynamics. [PERSON] et al. [32] integrated the model-based and data-driven methods to assess and control a power system's online frequency stability. [PERSON] et al. [33] proposed a hybrid physical model-based and data-driven approach for linearizing a power flow model. Although better prediction results could be obtained by the above methods, it also requires large training data. To address this issue, simulation data generated based on the mechanism model is always used to enhance the model training. [PERSON] et al. [34] proposed a mold temperature distribution prediction model which uses the simulation data and the simulation error as training data, while the model also needs training or adjustment during the manufacturing process. In another aspect, constructing a neural network based on a filtering algorithm is also a common fusion method. Input errors usually negatively affects the convergence of neural networks or lowers the effect of the model. Filtering algorithms are methods for removing data noise and restoring the original data. When the measurement variance is known, the Kalman filter can estimate the state of a dynamic system from a series of data with measurement noise. Therefore, the fusion of the Kalman filter algorithm and neural network can reduce the effect of input error. [PERSON] et al. [35] proposed a method to reduce the large roll motion of sailing ships which combines a Basis Function Neural Network and Unscented Kalman Filter to design a rudder roll stabilization system. [PERSON] et al. [36] proposed an extended Kalman filter-based artificial neural network method to eliminate the temperature effects and detect damages for structures equipped with long-term monitoring systems. [PERSON] et al. [37] proposed a fault detection and diagnosis method which employed a Kalman filter to estimate the condition of a blade pitch system and used an artificial neural network to diagnose a predetermined fault type. [PERSON] et al. [5] proposed a novel multi-sensor fusion algorithm for underwater vehicle localization that improves the state estimation by augmentation of the radial basis function neural network with an extended Kalman filter. [PERSON] et al. [4] proposed a method that applied a feed forward neural network and the extended Kalman filter to estimate the state of charge of the battery. [PERSON] et al. [38] proposed a calibration method for enhancing robot position accuracy using an extended Kalman filtering algorithm. The above existing research mainly targets data preprocessing and model training, while the Kalman filter can be used to narrow the distribution gap of the simulation data and real data, which have provided good inspiration to the authors. As already alluded to, deformation force is an important physical quantity associated with the deformation, and its monitoring can provide useful information for deformation control. Specific to aero-engine casings, due to their unique structural characteristics, the development of more advanced techniques of deformation force monitoring is called for. Therefore, in this paper, we propose a new deformation monitoring methodology specifically targeting parts like an aero-engine casing.