Energy-Consistent Neural Networks with Fenchel–Young Loss for Physics-Guided Energy Prediction in Sheet Metal Forming Under Small-Data Conditions

of 0.9778, corresponding to approximately a 48% reduction in RMSE compared to the best baseline model. Compared to an energy-constrained neural network without the Fenchel-Young term, prediction error is reduced by approximately 50% with substantially improved stability. These results indicate that embedding energy-consistent dual structures enhances both prediction accuracy and robustness, providing a practical surrogate modeling approach for process optimization in sheet metal forming under limited data availability.