PDE-Constrained Gaussian Process Model on Material Removal Rate of Wire Saw Slicing Process

Thickness uniformity of wafers is a critical quality measure in a wire saw slicing process. Nonuniformity occurs when the material removal rate (MRR) changes over time during a slicing process, and it poses a significant problem for the downstream processes such as lapping and polishing. Therefore, the MRR should be modeled and controlled to maintain the thickness uniformity. In this paper, a PDE-constrained Gaussian process model is developed based on the global Galerkin discretization of the governing partial differential equations (PDEs). Three features are incorporated into the statistical model: (1) the PDEs governing the wire saw slicing process, which are obtained from engineering knowledge, (2) the systematic errors of the manufacturing process, and (3) the random errors, including both random manufacturing errors and measurement noises. Real experiments are conducted to provide data for the validation of the PDE-constrained Gaussian process model by estimating the model coefficients and further using the model to predict the overall MRR profile. The results of cross-validation indicate that the prediction performance of the PDE-constrained Gaussian process model is better than the widely used universal Kriging model with a mean of second order polynomial functions.