Reliable nonlinear dynamic gray-box modeling by regularized training data estimation and sensitivity analysis

Dynamic process models are a key requirement for advanced process control and the application of process optimization techniques. The derivation of these models is time consuming and error-prone in cases where a lack of physico-chemical understanding is present. Machine learning (ML) methods can be employed in these cases to extract models or model elements from data. To reduce the amount of necessary data and to increase the extrapolation capabilities, gray-box models can be used that combine mechanistic equations with ML models. For embedded ML-models, the selection of a suitable model structure is challenging. Therefore, we propose a methodology to approach this problem in several steps by firstly estimating what values the ML-models should predict to accurately describe the experimental data. Subsequently, the ML-submodels can be trained using any ML-toolbox. Finally, a full parameter estimation is performed using a dynamic simulation in the cost function. We investigate different algorithmic options and show promising results for a case study of the fermentation of a sporulating bacterium.