BAT-NO: A Boundary-Condition-Aware Transformer Neural Operator for Crashworthiness Prediction of Vehicle Components

Haoran Li; Yingxue Zhao; Haosu Zhou; Mustapha Ziane; Pierre Culiere; Tobias Pfaff; Nan Li

cs.LGOct 2, 2026

BAT-NO: A Boundary-Condition-Aware Transformer Neural Operator for Crashworthiness Prediction of Vehicle Components

Authors: Haoran Li, Yingxue Zhao, Haosu Zhou, Mustapha Ziane, Pierre Culiere, Tobias Pfaff, Nan Li

Organizations: Dyson School of Design Engineering, Imperial College London, London, UK · Keysight Technologies, Bagneux, France · Prometheus, UK

Abstract

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.

Figures & tables

Figure 1: Overview of the proposed BAT-NO architecture.

Figure 2: FE side-impact setup and geometric morphing scheme used to generate Cases S, SL, and SLB. The illustration shows the B-pillar, impactor, connection regions, and the geometric control parameters used for shape variation.

Model	Case S	Case SL	Case SLB
MGN Pfaff et al. [2021]	1.8967	0.5925	2.1354
ReGUNet Li et al. [2026b]	0.1611	0.4356	1.1057
DeepONet Lu et al. [2021]	0.3239	0.3690	0.7607
FNO Li et al. [2021]	0.2181	0.2873	0.7772
GNO Li et al. [2020]	0.5555	0.3952	1.6343
GINO Li et al. [2023b]	0.3796	0.1752	0.6689

Table 1: Mean final-step MED over the 50 validation samples for the three case studies. Errors are reported in millimetres. The best result in each case is shown in bold, and the second-best result is underlined.

Figure 3: Qualitative baseline comparison of the accumulated $z$ -displacement field. For each model, the panels show the ground-truth field (GT), predicted field (Pred.), and signed prediction-error field for that model’s median-error validation sample. Displacement and error values are reported in millimetres.

Variant	Mean validation error (mm)
Base (GINO)	0.6689
Base + Slice	0.6139
Base + Slice + MP	0.5774
Base + Slice + MP + Rec.	0.5282
BAT-NO	0.4510

Table 2: Mean final-step MED over the 50 Case SLB validation simulations for the BAT-NO component ablation. Errors are reported in millimetres.

Figure 4: Visualisation of the learned node-to-slice assignments for a 16-slice example. Each panel corresponds to one slice, and the colour indicates the assignment score, representing how strongly each node is assigned to that slice.

Figure 5: Training and validation distributions of the final-step MED for the BAT-NO component ablation on Case SLB. The box plots show how prediction errors change as the slice mapping, pre-projection message passing (MP), recurrence block (Rec.), and complete BAT-NO architecture are introduced.

Figure 6: Mean nodal Euclidean displacement error over the autoregressive rollout for the BAT-NO component ablation on Case SLB. Dashed and solid lines denote the training and validation splits, respectively.

Scalar output	Global pooling	Grid query	Mesh query
Internal energy	5.40	5.04	5.29
Top spring force	4.53	4.21	3.89
Bottom-left spring force	5.10	4.25	3.90
Bottom-right spring force	4.95	4.85	4.83
Contact force	3.42	3.20	3.37
Impactor intrusion	3.17	2.97	2.99

Table 3: Mean relative trajectory errors for scalar-output prediction on the validation set and computational cost during training. Error values are percentages; lower is better. The best value in each row is shown in bold, and the second-best value is underlined.

Physical metric	Global pooling	Grid query	Mesh query
Top peak force	4.81	3.21	2.71
Bottom-left peak force	2.74	2.75	3.18
Bottom-right peak force	3.25	3.20	3.46
Contact peak force	2.43	2.41	2.73
Top impulse	2.91	2.27	2.40
Bottom-left impulse	3.30	2.37	2.43

Table 4: Median absolute relative errors in physical-response metrics on the validation set. Error values are percentages; lower is better. The best value in each row is shown in bold, and the second-best value is underlined.

Figure 7: Scalar-response prediction performance. (a) Time-resolved training-range NRMSE over the autoregressive rollout for the six scalar responses. (b) Validation distributions of the relative trajectory

L_{2}

error. (c) Ground-truth and predicted trajectories for the median-error validation sample. (d) Median absolute relative errors of the derived physical-response metrics, with error bars showing their variation across the validation samples.

Metric	ReGUNet	GINO	Transolver	BAT-NO
Internal energy	7.81	8.29	2.40	5.04
Top spring force	8.54	4.84	3.46	4.21
Bottom-left spring force	10.80	5.93	2.95	4.25
Bottom-right spring force	9.57	6.80	3.57	4.85
Contact force	5.46	5.09	2.02	3.20
Impactor intrusion	4.40	4.73	1.93	2.97

Table 5: Comparison of scalar-response and displacement-field rollout accuracy on the validation set. Scalar-response values are mean relative trajectory errors (%); field values are mean final-step MED values (mm). Lower is better. The best value in each row is shown in bold, and the second-best value is underlined. ReGUNet, GINO, and BAT-NO use the grid-query decoder, while Transolver uses the mesh-query decoder.

Panel of Figure 8 — Figure 8: Validation final-step MED for (a) the number of physics-attention slices and (b) the FNO depth.

Panel of Figure 9 — Figure 9: Accuracy–efficiency trade-off for different latent-grid resolutions: (a) validation error versus peak GPU memory and (b) validation error versus training time.

Model architecture		Training
Hyperparameter	Setting	Hyperparameter	Setting
Latent grid resolution	$20^{3}$	Learning rate	$3\times 10^{-4}$
Fourier modes	4	Batch size	2
FNO layers	3	Scalar-loss weight	0.10
Number of slices	16	Learning-rate schedule	Constant
Attention heads	4	Attention dropout	0

Table 6: Final hyperparameter settings used for BAT-NO.

Figure 10: Mean nodal Euclidean displacement error over the autoregressive rollout for the untuned field-only BAT-NO and the final tuned field-and-scalar BAT-NO. Training and validation curves are shown for both configurations, together with the tuned-model test error on

300

test samples.

Metric	Validation	Test
Internal energy	2.14	2.19
Top spring force	3.04	3.28
Bottom-left spring force	3.64	3.17
Bottom-right spring force	2.82	2.85
Contact force	1.77	1.84
Impactor intrusion	1.48	1.46

Table 7: Field and scalar-response accuracy of the final tuned BAT-NO under fully autoregressive evaluation. Scalar-response values are mean relative trajectory

L_{2}

errors (%).

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

	Shape range (%)			Loading range		Support stiffness range (kN/mm)
Case	$X$ morph	$Y$ morph	$Z$ morph	Impact position (mm)	Impact velocity (m/s)	Top	Bottom left	Bottom right
S	$[-5,5]$	$[0,10]$	$[-10,10]$	$90$	$4.0$	Fixed	Fixed	Fixed
SL	$[-2.5,2.5]$	$[0,5]$	$[-5,5]$	$[75,105]$	$[3.5,4.0]$	Fixed	Fixed	Fixed
SLB	$[-5,5]$	$[0,10]$	$[-10,10]$	$[75,105]$	$[3.5,4.0]$	$[1,10]$	$[1,10]$	$[1,10]$

Appendix

Table 8: Parameter ranges used to generate the three B-pillar datasets. Cases S and SL use fixed constraints at the support regions.

Figure 11: Box plots comparing the training and validation final-step MED values of the baseline models for Case S, in which only the B-pillar shape is varied.

Figure 12: Box plots comparing the training and validation final-step MED values of the baseline models for Case SL, in which the B-pillar shape and loading conditions are varied.

Figure 13: Box plots comparing the training and validation final-step MED values of the baseline models for Case SLB, in which the B-pillar shape, loading conditions, and three spring-based boundary conditions are varied.

Table 22

Table 23

Table 24

Table 25

Table 26

Table 27

Table 28

Panel of Figure 14 — Figure 14: Validation final-step MED for the first six BAT-NO hyperparameter screenings: (a) learning rate, (b) scalar-loss weight, (c) latent-grid resolution, (d) Fourier modes, (e) FNO depth, and (f) slice count.

Panel of Figure 15 — Figure 15: Validation final-step MED for the remaining BAT-NO hyperparameter screenings: (a) attention heads, (b) pre-grid message-passing steps, (c) slice-transformer depth, (d) attention dropout, (e) weight decay, and (f) maximum predicted-feedback probability.

Setting	Value
Optimiser	Adam
Learning rate	$3\times 10^{-4}$
Learning-rate schedule	Constant
Weight decay	$0$
Batch size	$2$
Training epochs	$1000$

Appendix

Table 23: Training and evaluation settings used for the final BAT-NO model.

Figure 16: Full-trajectory visualisation of the accumulated $z$ -displacement field for the median-error test sample. Each time step shows the ground-truth field, BAT-NO prediction, and prediction error. The error colour range is fixed to $\pm 5$ mm for all time steps.

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