Systematic tree search for symbolic regression : deterministically searching the space of dimensionally homogeneous models

Abstract

In engineering, the identification of the functional relationship between a set of physical variables is a common objective. In addition to physics, an increasing number of machine learning methods are being utilised to create these models. In particular, a machine learning algorithm, generally referred to as symbolic regression (SR), generates interpretable models in the form of mathematical functions. Thus, a physical interpretation of the relationship found remains possible. While the majority of contemporary SR algorithms do not employ a systematic approach, incorporating stochastic elements, this paper proposes a novel implementation of symbolic regression. This implementation takes the form of a systematic tree search, which is deterministically spanning and searching the space of possible symbolic models. For this purpose, new mathematical models are successively generated by small modifications of an existing model, whereby the search tree grows iteratively and with it the complexity of the models. The control of the search process, and thus the selection of the model to be changed is based on a specially designed dynamic heuristic. Through the additional use of dimensional analysis, the dimensional homogeneity of the models created can be guaranteed. The efficiency of the method to arrive at interpretable models from synthetic data is illustrated by finding each of the 12 Nguyen benchmark data sets. The robustness of the approach is shown by reconstructing Breguet’s range formula from data with varying degrees of noise.