The Road to Learning Explainable Inverse Kinematic Models: Graph Neural Networks as Inductive Bias for Symbolic Regression

This paper shows how a Graph Neural Network (GNN) can be used to learn an Inverse Kinematics (IK) based on an automatically generated dataset. The generated Inverse Kinematics is generalized to a family of manipulators with the same Degree of Freedom (DOF), but varying link length configurations. The results indicate a position error of less than 1.0 cm for 3 DOF and 4.5 cm for 5 DOF, and orientation error of 2$^\circ$ for 3 DOF and 8.2$^\circ$ for 6 DOF, which allows the deployment to certain real world-problems. However, out-of-domain errors and lack of extrapolation can be observed in the resulting GNN. An extensive analysis of these errors indicates potential for enhancement in the future. Consequently, the generated GNNs are tailored to be used in future work as an inductive bias to generate analytical equations through symbolic regression.

Shape Constraints in Symbolic Regression using Penalized Least Squares

We study the addition of shape constraints and their consideration during the parameter estimation step of symbolic regression (SR). Shape constraints serve as a means to introduce prior knowledge about the shape of the otherwise unknown model function into SR. Unlike previous works that have explored shape constraints in SR, we propose minimizing shape constraint violations during parameter estimation using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying three symbolic expressions from a synthetically generated data set. This paper examines two benchmark scenarios: one with varying noise levels and another with reduced amounts of training data. The results indicate that incorporating shape constraints into the expression search is particularly beneficial when data is scarce. Compared to using shape constraints only in the selection process, our approach of minimizing violations during parameter estimation shows a statistically significant benefit in some of our test cases, without being significantly worse in any instance.

Unit-Aware Genetic Programming for the Development of Empirical Equations

When developing empirical equations, domain experts require these to be accurate and adhere to physical laws. Often, constants with unknown units need to be discovered alongside the equations. Traditional unit-aware genetic programming (GP) approaches cannot be used when unknown constants with undetermined units are included. This paper presents a method for dimensional analysis that propagates unknown units as ''jokers'' and returns the magnitude of unit violations. We propose three methods, namely evolutive culling, a repair mechanism, and a multi-objective approach, to integrate the dimensional analysis in the GP algorithm. Experiments on datasets with ground truth demonstrate comparable performance of evolutive culling and the multi-objective approach to a baseline without dimensional analysis. Extensive analysis of the results on datasets without ground truth reveals that the unit-aware algorithms make only low sacrifices in accuracy, while producing unit-adherent solutions. Overall, we presented a promising novel approach for developing unit-adherent empirical equations.