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.

In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation, then the solution mapping is regarded as a continuous metric-to-metric mapping, and subsequently can be represented by another continuous metric-to-Banach mapping using two different strategies, referred to as the D2D framework and the D2E framework, respectively. We point out that such a metric-to-Banach mapping can be learned by neural networks, hence the solution mapping is accordingly learned. With this framework, a rigorous convergence analysis is built for the problem of learning solution mappings of PDEs on varying domains. As the theoretical framework holds based on several pivotal assumptions which need to be verified for a given specific problem, we study the star domains as a typical example, and other situations could be similarly verified. There are three important features of this framework: (1) The domains under consideration are not required to be diffeomorphic, therefore a wide range of regions can be covered by one model provided they are homeomorphic. (2) The deformation mapping is unnecessary to be continuous, thus it can be flexibly established via combining a primary identity mapping and a local deformation mapping. This capability facilitates the resolution of large systems where only local parts of the geometry undergo change. (3) If a linearity-preserving neural operator such as MIONet is adopted, this framework still preserves the linearity of the surrogate solution mapping on its source term for linear PDEs, thus it can be applied to the hybrid iterative method. We finally present several numerical experiments to validate our theoretical results.

Real-time monitoring of critical parameters is essential for energy systems' safe and efficient operation. However, traditional sensors often fail and degrade in harsh environments where physical sensors cannot be placed (inaccessible locations). In addition, there are important parameters that cannot be directly measured by sensors. We need machine learning (ML)-based real-time monitoring in those remote locations to ensure system operations. However, traditional ML models struggle to process continuous sensor profile data to fit model requirements, leading to the loss of spatial relationships. Another challenge for real-time monitoring is ``dataset shift" and the need for frequent retraining under varying conditions, where extensive retraining prohibits real-time inference. To resolve these challenges, this study addressed the limitations of real-time monitoring methods by enabling monitoring in locations where physical sensors are impractical to deploy. Our proposed approach, utilizing Multi-Input Operator Network virtual sensors, leverages deep learning to seamlessly integrate diverse data sources and accurately predict key parameters in real-time without the need for additional physical sensors. The approach's effectiveness is demonstrated through thermal-hydraulic monitoring in a nuclear reactor subchannel, achieving remarkable accuracy.

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces, establishing that MIONet can approximate mappings with multiple inputs in metric spaces. Subsequently, we construct a set consisting of some appropriate regions and provide a metric on this set thus make it a metric space, which satisfies the approximation condition of MIONet. Building upon the theoretical foundation, we are able to learn the solution mapping of a PDE with all the parameters varying, including the parameters of the differential operator, the right-hand side term, the boundary condition, as well as the domain. Without loss of generality, we for example perform the experiments for 2-d Poisson equations, where the domains and the right-hand side terms are varying. The results provide insights into the performance of this method across convex polygons, polar regions with smooth boundary, and predictions for different levels of discretization on one task. We also show the additional result of the fully-parameterized case in the appendix for interested readers. Reasonably, we point out that this is a meshless method, hence can be flexibly used as a general solver for a type of PDE.

We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.

Rapid surrogate models derived from observational data now substantially reduce the computational cost to solve practical problems like solid mechanics [1], structural health monitoring [2, 3, 4], field problem solutions [5], fault diagnosis [6, 7], medical imaging [8, 9], autonomous driving [10], and power grid simulation [11] A significant challenge in current neural network surrogate models lies in their generalization capability. Addressing this, the foundational work [12] introduced Operator Learning, a novel method aimed at learning the mapping between different function spaces. Building on this, [13] developed the Deep Operator Neural Network (DeepONet), capable of being trained with limited datasets while minimizing generalization errors. This influential research has been applied in various domains, including the prediction of linear instability waves in high-speed boundary layers [14], forecasting power grid's post-fault trajectories [15], learning nonlinear operators in oscillatory function spaces for seismic wave responses [4], and analyzing nanoscale heat transport [16]. Additionally, several advancements of DeepONet have been proposed, such as Bayesian DeepONet [17, 18], DeepONet with proper orthogonal decomposition [19], multiscale DeepONet [4], a neural operator with coupled attention [20], physics-informed DeepONet [21, 22], and the multiple-input deep neural operators (MIONet) [23].