Physics-informed DeepONet with stiffness-based loss functions for structural response prediction

TL;DR

This work develops a physics-informed DeepONet framework that leverages stiffness-based energy conservation and static-equilibrium losses to predict full-field structural responses under varied static loads. By comparing output-management strategies (N-independent DeepONets vs. split branch/trunk) and introducing Schur-complement-based SE-S losses, the approach achieves real-time predictions with high accuracy while dramatically reducing training time. Validation on a 2D beam and the KW-51 bridge demonstrates strong generalization to unseen loading and significant computational gains, suggesting practical applicability for rapid FE-model augmentation and real-time monitoring. The methodology offers a pathway to extend to nonlinear dynamics and time-dependent effects, enabling scalable, physics-consistent operator learning for complex civil structures.

Abstract

Finite element modeling is a well-established tool for structural analysis, yet modeling complex structures often requires extensive pre-processing, significant analysis effort, and considerable time. This study addresses this challenge by introducing an innovative method for real-time prediction of structural static responses using DeepOnet which relies on a novel approach to physics-informed networks driven by structural balance laws. This approach offers the flexibility to accurately predict responses under various load classes and magnitudes. The trained DeepONet can generate solutions for the entire domain, within a fraction of a second. This capability effectively eliminates the need for extensive remodeling and analysis typically required for each new case in FE modeling. We apply the proposed method to two structures: a simple 2D beam structure and a comprehensive 3D model of a real bridge. To predict multiple variables with DeepONet, we utilize two strategies: a split branch/trunk and multiple DeepONets combined into a single DeepONet. In addition to data-driven training, we introduce a novel physics-informed training approaches. This method leverages structural stiffness matrices to enforce fundamental equilibrium and energy conservation principles, resulting in two novel physics-informed loss functions: energy conservation and static equilibrium using the Schur complement. We use various combinations of loss functions to achieve an error rate of less than 5% with significantly reduced training time. This study shows that DeepONet, enhanced with hybrid loss functions, can accurately and efficiently predict displacements and rotations at each mesh point, with reduced training time.

Figures (20)

• Figure 1: Flowchart of the proposed methodology
• Figure 2: Illustration of DeepONet structure and types. A. Stacked DeepONet, B. Unstacked DeepONet, C. Multilayer Perceptrons(MLP) architecture of Branch/Trunk Network
• Figure 3: Illustration to handle multiple outputs using DeepONet: A. 6 Independent DeepONets to handle 6 outputs, B. 1 DeepONet to handle 3 outputs.
• Figure 4: Illustration of the Schur Complement to reduce the size of the system: A. Total nodes in the structural domain, B. Picked nodes by applying Schur complement, C. The remaining nodes solution obtained by using post-processing (Eq. \ref{['Matrix_2']})
• Figure 5: Illustration of the 2D beam structure and loading scenarios: A. FEM nodes, B. 1st loading scenario: UDL on half base, C. 2nd loading scenario: UVL on half base, D. 3rd loading scenario: UDL on full base