Stochastic production system (SPS) refers to a production process that is influenced by a large number of random factors, typical examples including industrial biosystem, composite material production system, and batch chemical reaction system. Notably, SPS is notorious for significant uncertainty and stochasticity, thereby making implementing process monitoring to ensure product quality a daunting task. One of the major underlying obstacles is how to accurately detect anomalies thereof in real time. To resolve so, this paper proposes a deep Koopman neural network-based approach, wherein two deep neural networks constitute an objective mapping between original data space and a linear high-dimensional space, and a linear operator describes dynamic evolution in the linear space. The performance of the proposed method is tested on two examples of SPS, which are of significant intrinsic stochastic dynamics, hence arguably constituting a novel class of benchmarks for performance comparing of various process monitoring algorithms, and becoming another contribution of this paper.