BiLoGo: Reducing the smoothing of localised responses in Transformer-based neural operators for sheet forming

Explicit dynamic finite element (FE) simulations are widely used for large-deformation analysis but remain expensive for design-space exploration. In explicit FE analysis, nodal kinematics and element-level deformation measures evolve through a coupled node-element update process. An FE-inspired learned simulator can preserve this structure and learn nodal kinematics and elemental deformation on their native domains. Nodal displacement describes geometry evolution, while elemental thinning identifies local deformation and critical regions. Feeding both outputs back during rollout allows each field to inform subsequent updates of the other. This work proposes CAtt-BiGNN, a cross-attention-based bipartite graph neural network for coupled nodal-elemental learning. The bipartite graph encodes FE-guided state organisation and information flow without imposing equilibrium or constitutive constraints. Mesh nodes and elements are represented as distinct graph entities connected by directed node-element edges, enabling prediction of nodal displacement increments and elemental deformation on their native domains. An edge-aware cross-attention processor dynamically updates edge representations and weights messages using the receiving state and evolving local relations. For larger graphs, CAtt-BiUGNN uses separate nodal and elemental downsampling and upsampling to enhance long-range propagation while preserving their interaction across graph levels. The method is evaluated on dome-shaped cold-forming and corner-shaped hot-forming benchmarks. Comparisons with node-centred baselines, an FE-inspired hypergraph model, a Physics-Attention-based neural operator, a graph Transformer, and attention ablations show improved joint accuracy in nodal displacement and elemental thinning prediction during autoregressive rollout.

High-fidelity finite-element simulations provide accurate crashworthiness predictions, but their cost limits iterative design exploration. Deep learning surrogates can reduce this cost, but many component-level models are developed under a single prescribed boundary condition, limiting generalisation to boundary variations. This work proposes a Boundary-Condition-Aware Transformer Neural Operator (BAT-NO) for autoregressive prediction of transient displacement fields and scalar crashworthiness responses under variations in geometry and boundary conditions. A B-pillar simulation framework evaluates generalisation across variations in geometry, impact position and velocity, and support stiffness. BAT-NO combines recurrent mesh processing with latent-grid Fourier operator processing. Boundary-condition information is transferred to the latent grid through a hybrid local--global mechanism. Slice-based attention models interactions among physically related regions, while direct boundary-to-grid projection preserves local spatial structure. Across the validation sets for the shape-only, shape-and-loading, and shape-loading-boundary cases, BAT-NO achieves the lowest mean final-step mean nodal Euclidean displacement error among the evaluated baselines. In the most challenging case, it reduces the mean error by 32.6% relative to the second-best model. Hyperparameter tuning reduces the validation error from 0.451 to 0.269 mm, with a comparable error of 0.267 mm on 300 unseen test simulations sampled within the investigated design space. An attention-based scalar decoder jointly predicts six response trajectories with a mean relative error of 2.46%. Most derived crashworthiness indicators have median errors below 3%. These results show that explicit local and global boundary-condition representations improve crashworthiness prediction over expanded component-level design spaces.

Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative $L_2$ error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.