  The modeling of multistage manufacturing systems (MMSs) has attracted
increased attention from both academia and industry. Recent advancements in
deep learning methods provide an opportunity to accomplish this task with
reduced cost and expertise. This study introduces a stochastic deep Koopman
(SDK) framework to model the complex behavior of MMSs. Specifically, we present
a novel application of Koopman operators to propagate critical quality
information extracted by variational autoencoders. Through this framework, we
can effectively capture the general nonlinear evolution of product quality
using a transferred linear representation, thus enhancing the interpretability
of the data-driven model. To evaluate the performance of the SDK framework, we
carried out a comparative study on an open-source dataset. The main findings of
this paper are as follows. Our results indicate that SDK surpasses other
popular data-driven models in accuracy when predicting stagewise product
quality within the MMS. Furthermore, the unique linear propagation property in
the stochastic latent space of SDK enables traceability for quality evolution
throughout the process, thereby facilitating the design of root cause analysis
schemes. Notably, the proposed framework requires minimal knowledge of the
underlying physics of production lines. It serves as a virtual metrology tool
that can be applied to various MMSs, contributing to the ultimate goal of Zero
Defect Manufacturing.
