From residual learning to explicit correlation revision: An interpretable machine learning framework for cross-fuel spray penetration prediction

Abstract

Predicting macroscopic spray characteristics is essential for optimizing fuel injection. To overcome the limited accuracy of conventional empirical correlations for liquid-phase penetration under high-temperature, high-pressure and cross-fuel conditions, this study proposes an interpretable machine-learning-assisted explicit correction method. A diesel/methanol spray database comprising 718 valid samples was established based on optical spray experiments conducted in this study. Using the Hiroyasu-Arai equation as the physical baseline, logarithmic relative residuals were defined and modeled using Light Gradient Boosting Machine (LightGBM). SHapley Additive exPlanations (SHAP) were then applied to identify dominant error sources and reconstruct nonlinear deviations into explicit correction terms. Results show that increasing ambient temperature suppresses liquid-phase penetration, with methanol exhibiting stronger thermal sensitivity. LightGBM outperformed eXtreme Gradient Boosting (XGBoost) and Categorical Boosting (CatBoost) in predictive accuracy and stability, while SHAP effectively revealed the origins of structural residuals. Compared with the original Hiroyasu-Arai correlation (R² = −1.3298), the corrected correlation achieved an R² of 0.9486, together with substantial reductions in Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The proposed closed-loop framework integrates experiments, baseline evaluation, residual learning, SHAP attribution, and explicit reconstruction, demonstrating an effective route for diagnosing and refining empirical spray correlations under complex conditions.