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