Evolutionary neural networks for learning turbulence closure models with explicit expressions

Developing physical closure models with explicit expressions based on a given dataset is essential to science and engineering. For such symbolic regression tasks, biology-inspired evolutionary algorithms are most widely used. However, typical evolutionary algorithms do not utilize any structural information inherent in training data, which limits their performance in finding accurate model structures and coefficients. By combining one evolutionary algorithm, gene expression programing (GEP), with an artificial neural network (ANN) for symbolic regression, we propose a novel evolutionary neural network method, in which candidate expressions are specifically designed so that they can be transformed between the GEP and ANN structures during training iterations. By combining the GEP's global searching and the ANN's gradient optimization capabilities, efficient and robust convergence to accurate models can be achieved. In addition, sparsity-enhancing strategies have been introduced to improve the interpretability of the trained models. The present method has been tested for finding different physical laws and then applied to turbulence modeling problems with different configurations, showing advantages compared to the existing GEP and ANN methods.