A framework for learning symbolic turbulence models from indirect observation data via neural networks and feature importance analysis

TL;DR

This paper introduces a neural network-based framework that efficiently learns symbolic turbulence models from indirect data, combining gradient-based training with feature importance analysis for improved interpretability and convergence.

Contribution

It presents a novel gradient-based approach integrating neural networks and feature importance analysis to learn symbolic turbulence models from indirect observations.

Findings

Successfully learned analytic models from indirect velocity data in duct flow.

Significantly improved training efficiency with feature importance analysis.

Achieved satisfactory generalizability of the learned turbulence models.

Abstract

Learning symbolic turbulence models from indirect observation data is of significant interest as it not only improves the accuracy of posterior prediction but also provides explicit model formulations with good interpretability. However, it typically resorts to gradient-free evolutionary algorithms, which can be relatively inefficient compared to gradient-based approaches, particularly when the Reynolds-averaged Navier-Stokes (RANS) simulations are involved in the training process. In view of this difficulty, we propose a framework that uses neural networks and the associated feature importance analysis to improve the efficiency of symbolic turbulence modeling. In doing so, the gradient-based method can be used to efficiently learn neural network-based representations of Reynolds stress from indirect data, which is further transformed into simplified mathematical expressions with symbolic regression. Moreover, feature importance analysis is introduced to accelerate the convergence of symbolic regression by excluding insignificant input features. The proposed training strategy is tested in the flow in a square duct, where it correctly learns underlying analytic models from indirect velocity data. Further, the method is applied in the flow over the periodic hills, demonstrating that the feature importance analysis can significantly improve the training efficiency and learn symbolic turbulence models with satisfactory generalizability.