Mostafa E. Mobasher

Bilal Ahmed, Yuqing Qiu, Diab W. Abueidda et al.

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.

Panos Pantidis, Habiba Eldababy, Christopher Miguel Tagle et al.

In our recently proposed Integrated Finite Element Neural Network (I-FENN) framework (Pantidis and Mobasher, 2023) we showcased how PINNs can be deployed on a finite element-level basis to swiftly approximate a state variable of interest, and we applied it in the context of non-local gradient-enhanced damage mechanics. In this paper, we enhance the rigour and performance of I-FENN by focusing on two crucial aspects of its PINN component: a) the error convergence analysis and b) the hyperparameter-performance relationship. Guided by the available theoretical formulations in the field, we introduce a systematic numerical approach based on a novel set of holistic performance metrics to answer both objectives. In the first objective, we explore in detail the convergence of the PINN training error and the global error against the network size and the training sample size. We demonstrate a consistent converging behavior of the two error types for any investigated combination of network complexity, dataset size and choice of hyperparameters, which empirically proves the conformance of the PINN setup and implementation to the available convergence theories. In the second objective, we establish an a-priori knowledge of the hyperparameters which favor higher predictive accuracy, lower computational effort, and the least chances of arriving at trivial solutions. The analysis leads to several outcomes that contribute to the better performance of I-FENN, and fills a long-standing gap in the PINN literature with regards to the numerical convergence of the network errors while accounting for commonly used optimizers (Adam and L-BFGS). The proposed analysis method can be directly extended to other ML applications in science and engineering. The code and data utilized in the analysis are posted publicly to aid the reproduction and extension of this research.

Bilal Ahmed, Yuqing Qiu, Diab W. Abueidda et al.

Finite element (FE) modeling is essential for structural analysis but remains computationally intensive, especially under dynamic loading. While operator learning models have shown promise in replicating static structural responses at FEM level accuracy, modeling dynamic behavior remains more challenging. This work presents a Multiple Input Operator Network (MIONet) that incorporates a second trunk network to explicitly encode temporal dynamics, enabling accurate prediction of structural responses under moving loads. Traditional DeepONet architectures using recurrent neural networks (RNNs) are limited by fixed time discretization and struggle to capture continuous dynamics. In contrast, MIONet predicts responses continuously over both space and time, removing the need for step wise modeling. It maps scalar inputs including load type, velocity, spatial mesh, and time steps to full field structural responses. To improve efficiency and enforce physical consistency, we introduce a physics informed loss based on dynamic equilibrium using precomputed mass, damping, and stiffness matrices, without solving the governing PDEs directly. Further, a Schur complement formulation reduces the training domain, significantly cutting computational costs while preserving global accuracy. The model is validated on both a simple beam and the KW-51 bridge, achieving FEM level accuracy within seconds. Compared to GRU based DeepONet, our model offers comparable accuracy with improved temporal continuity and over 100 times faster inference, making it well suited for real-time structural monitoring and digital twin applications.