Hybrid models combining mechanistic models with adaptive regression splines and local stepwise regression

This paper introduces a hybrid modeling approach based on the combination of prior knowledge, under the form of mechanistic models, with tools devoted to the extraction of knowledge from operating data. The first module captures first-principles system behavior, whereas the second models output residuals between real data and mechanistic predictions. The empirical module comprises two sequential tools: one to partition the time domain into zones where residuals are well-fitted by time-dependent univariate piecewise linear polynomials and the other to fit local regression models of residuals within such zones, considering process inputs and mechanistic predictions as independent variables. Time domain partitioning is carried out by an adaptive regression splines (ARS) approach, whereas the construction of time-discrete regression models for each of the different zones thus identified is achieved by stepwise regression (SR). The quality of time domain partitioning and the performance of this hybrid modeling approach are evaluated with data obtained from a simulated fed-batch penicillin fermentation process. The ARS module adequately captures the zones where system behavior presents explicit time-dependent trends, and the SR module produces local models with good interpretability. The results obtained show that our hybrid approach outperforms other techniques applied to the same problem, particularly when local regression models include autoregressive terms.