Energy-Consistent Neural Networks with Fenchel-Young Loss for Physics-Guided Energy Prediction in Sheet Metal Forming...

This study addresses energy-response prediction in sheet metal forming under small-data conditions, where conventional simulation-based approaches are computationally expensive and data acquisition is limited. We propose an Energy-Informed Neural Network (EINN) framework that integrates energy consistency constraints and a Fenchel-Young duality-based loss to enforce physically consistent learning without relying on explicit governing equations. Using a dataset generated from 54 finite element simulations across 18 materials and three friction conditions, the proposed model demonstrates significant performance improvements. Specifically, EINN achieves an RMSE of 0.0096, MAE of 0.0065, and R² 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.