We investigate the addition of constraints on the function image and its derivatives for the incorporation of prior knowledge in symbolic regression. The approach is called shape-constrained symbolic regression and allows us to enforce, for example, monotonicity of the function over selected inputs. The aim is to find models which conform to expected behavior and which have improved extrapolation capabilities. We demonstrate the feasibility of the idea and propose and compare two evolutionary algorithms for shape-constrained symbolic regression: (i) an extension of tree-based genetic programming which discards infeasible solutions in the selection step, and (ii) a two-population evolutionary algorithm that separates the feasible from the infeasible solutions. In both algorithms we use interval arithmetic to approximate bounds for models and their partial derivatives. The algorithms are tested on a set of 19 synthetic and four real-world regression problems. Both algorithms are able to identify models which conform to shape constraints which is not the case for the unmodified symbolic regression algorithms. However, the predictive accuracy of models with constraints is worse on the training set and the test set. Shape-constrained polynomial regression produces the best results for the test set but also significantly larger models.

In this publication a constant optimization approach for symbolic regression is introduced to separate the task of finding the correct model structure from the necessity to evolve the correct numerical constants. A gradient-based nonlinear least squares optimization algorithm, the Levenberg-Marquardt (LM) algorithm, is used for adjusting constant values in symbolic expression trees during their evolution. The LM algorithm depends on gradient information consisting of partial derivations of the trees, which are obtained by automatic differentiation. The presented constant optimization approach is tested on several benchmark problems and compared to a standard genetic programming algorithm to show its effectiveness. Although the constant optimization involves an overhead regarding the execution time, the achieved accuracy increases significantly as well as the ability of genetic programming to learn from provided data. As an example, the Pagie-1 problem could be solved in 37 out of 50 test runs, whereas without constant optimization it was solved in only 10 runs. Furthermore, different configurations of the constant optimization approach (number of iterations, probability of applying constant optimization) are evaluated and their impact is detailed in the results section.

This paper investigates the modeling of brushless permanent-magnet synchronous machines (PMSMs). The focus is on deriving an automatable process for obtaining dynamic motor models that take nonlinear effects, such as saturation, into account. The modeling is based on finite element (FE) simulations for different current vectors in the$dq plane over a full electrical period. The parameters obtained are the stator flux in terms of the direct and quadrature components and the air-gap torque, both modeled as functions of the rotor angle and the current vector. The data are preprocessed according to theoretical results on potential harmonics in the targets as functions of the rotor angle. A variety of modeling strategies were explored: linear regression, support vector machines, symbolic regression using genetic programming, random forests, and artificial neural networks. The motor models were optimized for each training technique, and their accuracy was then compared based on the initially available FE data and further FE simulations for additional current vectors. Artificial neural networks and symbolic regression using genetic programming achieved the highest accuracy, particularly with additional test data.