Integration of Physical Knowledge in Empirical Models - A New Approach to Regression Analysis

Design of experiments, empirical modelling and model-based optimization is a widely known and approved approach for high-dimensional optimization problems in powertrain engineering. A main drawback of many empirical modelling algorithms is the undetermined extrapolation behaviour. Besides that, those black box models must strictly rely on the provided measurement data. Neither quantitative nor qualitative physical knowledge can typically be integrated in those models easily. We discuss a general approach called shape-constrained regression for the integration of qualitative physical knowledge in empirical modelling algorithms. For example, this allows capturing knowledge such as monotonicity over selected inputs (e.g. “the dependent variable increases when input x is increased”). The exact functional relationship between dependent variables and the independent variables is unknown, can be affected by interactions, and must be identified by the modelling algorithm. We demonstrate the effectivity of the approach by applying polynomial regression with shape constraints for modelling emissions. Using this approach, we find polynomial models, which conform to physical knowledge and extrapolate well within the specified bounds. The shape-constrained models have slightly higher fitting errors. The process of fitting with shape-constraints takes approximately one to two minutes on a desktop machine for each polynomial and depends mainly on the number of constraints.