Organizations: Institute of Computational Mechanics × AI & College of Intelligent Robotics and Advanced Manufacturing, Fudan University, Shanghai 200433, China · School of Aerospace Engineering, Xiamen University, Xiamen 361102, China · Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China · Institute of Photonics, Department of Mathematics and Physics, Leibniz University, Hannover, Germany · Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstr. 15, D-99423, Weimar, Germany
Plate and shell structures are widely used in engineering, making rapid response prediction under varying geometries, materials, and loads highly desirable. However, conventional finite element methods require repeated modeling and solution, resulting in high computational costs. This study proposes a geometry-aware variational neural operator for Mindlin-Reissner plate problems, termed MR-GVNO. The method uses boundary point clouds to represent irregular geometries and employs separate encoders for spatially varying material fields, pressure loads, and scalar physical parameters. A cross-attention mechanism integrates these inputs with query point information to predict transverse deflections and rotations at arbitrary locations. MR-GVNO is trained without labeled solution data using a variational physics-informed loss derived from the discretized total potential energy. It directly processes irregular point clouds and allows different physical fields to be discretized independently, avoiding interpolation onto a common grid. Numerical experiments on single-hole, double-hole, and L-shaped plates demonstrate accurate response prediction under homogeneous and heterogeneous materials and uniform and random loads. The model also achieves millisecond-level full-field inference and favorable cross-geometry generalization.
Finite element analysis (FEA) is essential for structural design but remains computationally expensive, particularly when evaluating multiple design iterations or load scenarios. Machine learning surrogate models offer a promising alternative, yet most approaches struggle with a critical limitation: generalizing across varying geometries. This work presents a mesh graph network (MGN) for predicting von Mises stress fields in 2D structural components with arbitrary hole geometries. Unlike traditional machine learning approaches that use absolute node coordinates as features, the proposed model builds on existing MGN frameworks that encode node types (e.g., fixed boundary, free surface, hole edge), relative edge features (distance between neighbors), and global features (applied load). This architecture is inherently translation- and rotation-invariant, enabling generalization to unseen geometries without retraining. The MGN was trained on 11 plate geometries under 20 load conditions and evaluated on 7 unseen geometries and 3 unseen loads. In the most favorable case, the model achieves R2≥0.97 on an unseen geometry and unseen load, compared to R2≈0.01--0.86 for conventional models (Random Forest, Gradient Boosting , K-Nearest Neighbors) trained on identical data. However, even in less favorable cases, the MGN model still outperforms conventional models. This work extends the mesh-based simulation framework of Pfaff et al. (arXiv:2010.03409) to structural mechanics, demonstrating that graph neural networks can serve as efficient surrogates for finite element analysis across varying geometries.
We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the φ-finite element method (φ-FEM). φ-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function φ. To impose the boundary conditions, Dirichlet problems adopt the φ-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with φ-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear φ-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80% compared with purely data-driven methods.
Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO