Learning accurate numerical constants when developing algebraic models is a
known challenge for evolutionary algorithms, such as Gene Expression
Programming (GEP). This paper introduces the concept of adaptive symbols to the
GEP framework by Weatheritt and Sandberg (2016) to develop advanced physics
closure models. Adaptive symbols utilize gradient information to learn locally
optimal numerical constants during model training, for which we investigate two
types of nonlinear optimization algorithms. The second contribution of this
work is implementing two regularization techniques to incentivize the
development of implementable and interpretable closure models. We apply L2​
regularization to ensure small magnitude numerical constants and devise a novel
complexity metric that supports the development of low complexity models via
custom symbol complexities and multi-objective optimization. This extended
framework is employed to four use cases, namely rediscovering Sutherland's
viscosity law, developing laminar flame speed combustion models and training
two types of fluid dynamics turbulence models. The model prediction accuracy
and the convergence speed of training are improved significantly across all of
the more and less complex use cases, respectively. The two regularization
methods are essential for developing implementable closure models and we
demonstrate that the developed turbulence models substantially improve
simulations over state-of-the-art models