Data-Driven Process Optimization Considering Surrogate Model Prediction Uncertainty: A Mixture Density Network-Based Approach

The artificial neural network (ANN) can be effectively used as a data-driven surrogate model in process optimization. However, there is a problem that the change of training set leads to prediction uncertainty. A novel framework is proposed in this paper to address this issue. In the proposed approach, an ensemble of ReLU ANNs is first trained with different training sets to simulate the prediction uncertainty caused by the training set variation. Then, a mixture density network (MDN) is used to approximate the ReLU ANN ensemble and it is further embedded into a mixed-integer linear optimization problem. The original optimization problem is reformulated into a chance-constrained form with the mean-variance-type objective function to address both constraint and objective uncertainties. The proposed approach is applied to a numerical example and two case studies to show its capability of solving complex process optimization problems under the neural network model prediction uncertainty.