Physics-informed deep operator networks with stiffness-based loss functions for structural response prediction

Finite element (FE) modeling is a powerful tool for structural analysis, but it often involves extensive pre-processing, significant analysis efforts, and time-consuming computations, especially for complex structures. To overcome these challenges, this study presents an innovative approach for real-time prediction of structural static responses using Deep Operator Networks (DeepONet). This method leverages a novel, physics-informed network guided by structural balance laws, enabling accurate predictions for various load scenarios. The trained DeepONet can generate solutions for the entire domain across every mesh point in a fraction of a second, eliminating the need for repetitive FE modeling for each new case. The proposed method is applied to two structures: a simple beam and a real-life model of the KW-51 bridge. To predict multiple variables, two strategies are employed: a split branch/trunk approach and multiple DeepONets combined into one. A parametric study optimizes the network's design, considering factors like neurons, layers, batch size, and aspect ratio, ensuring high accuracy while avoiding underfitting and overfitting. Beyond data-driven (DD) training, the study introduces novel physics-informed training methods that utilize structural stiffness matrices to enforce equilibrium and energy conservation principles. These methods lead to two new loss functions: energy conservation (EC) and static equilibrium using the Schur complement (SE-S). By combining these loss functions, the model achieves less than 5% error with significantly reduced training time. This study shows that DeepONet, enhanced with hybrid loss functions, can efficiently and accurately predict structural responses at each mesh point, with minimal training time.