Field Inversion Symbolic Regression with Embedded Equation Learner for Interpretable Turbulence Model Correction

Abstract

An interpretable, physics-consistent turbulence model correction framework, termed FISR-Equation Learner (EQL), is proposed by embedding equation learning directly into a Partial Differential Equations (PDE)-constrained field inversion process based on the adjoint method. Unlike conventional two-stage approaches, the correction model is optimized end-to-end in parameter space using an EQL architecture, enabling the direct identification of compact analytical expressions while maintaining consistency with the governing equations. The method is applied to the shear-stress-transport (SST) model and trained on two canonical separated flows, the curved backward-facing step and the NASA hump. The resulting explicit expression significantly reduces separation bubble overprediction and improves reattachment prediction, achieving performance comparable to neural-network-based end-to-end methods while retaining full interpretability. Generalization is demonstrated on unseen configurations, including periodic hills, a surface-mounted cube, and the high-lift NLR7301 airfoil. The model improves separated-flow predictions and stall characteristics without degrading attached boundary-layer performance. Overall, FISR-EQL provides a practical pathway toward optimal yet transparent data-driven turbulence model correction.