HEM-constrained PINN with SHAP-guided optimization for predicting sCO2 critical flow through orifices across broad parameter conditions

Accurate and efficient prediction of critical mass flow rates for supercritical carbon dioxide (sCO₂) system is essential for real-time risk assessment and safety management. However, current efficient models exhibit limited generalizability due to data scarcity across broad parameter conditions. To address this, a homogeneous equilibrium model-constrained physics-informed neural network is proposed, using sCO₂ critical flow through orifices as a case study. The model incorporates fundamental physical mechanisms into the network’s cost function, enabling compliance with physical laws and enhancing generalizability with limited training data. Furthermore, a Shapley additive explanation (SHAP)-guided optimization method is introduced to mitigate the prevalent “black-box” limitation. This approach integrates prior physical knowledge with derivative information of mean SHAP curves derived from interpretability analysis, refining the weights of the network’s cost function to enhance the consistency between the model’s behavior and the fundamental principles of thermodynamics and fluid mechanics, thereby improving interpretability. Validated against experimental and simulation datasets spanning broad parameter conditions, the proposed model achieves a 62.65% reduction in mean relative error compared to the existing model, demonstrating superior accuracy and generalization. Subsequent SHAP-guided optimization further reduces the error by 15.81%, outperforming validation-set-error-based tuning while ensuring input feature contributions better adhered to physical laws. This study can provide theoretical reference for the accurate, efficient and interpretable prediction of sCO₂ critical mass flow rates.