CoPiT: A conditional Position-induced Transformer for aerodynamic field prediction on unstructured meshes

Motivation

• Existing aerodynamic surrogates face geometry-dependent meshes and global conditions, while input concatenation leaves condition effects on intermediate representations implicit.Whether intermediate conditioning improves field and integrated-load prediction, and how placement affects metrics, remain empirical questions.Source: §1 · p. 2

Evaluation

• CoPiT is evaluated on two-dimensional AirfRANS and three-dimensional BlendedNet++, including scarce-data and extrapolation settings on AirfRANS.The evaluation includes two-dimensional airfoil fields and three-dimensional blended-wing-body surface fields.Source: §1 · pp. 2–3

Architecture

• CoPiT combines PiT’s geometry-based position-attention with multi-stage FiLM that modulates intermediate features using global parameters.FiLM is applied in the processor and decoder, without altering geometry-based attention weights.Source: §1 · pp. 2–3

Analysis

• Under FiLM-only conditioning, encoder features and attention weights are condition-independent for fixed geometry, and predictions are Lipschitz continuous in the condition.The paper also shows that pointwise FiLM preserves convergence of an already convergent attention output under bounded modulation.Source: §1 · p. 3

Evaluation

• On AirfRANS, CoPiT reduces surface and volume losses by approximately 54% and 24% relative to Transolver, respectively, and has the lowest overall normalized test MSE among reimplemented BlendedNet++ baselines.Matched ablations find FiLM generally outperforms input concatenation, while preferred FiLM placement depends on the evaluation metric.Source: §1 · p. 3

Neural operators

• Neural-operator approaches include DeepONet, FNO, geometry extensions Geo-FNO and GINO, graph networks, and transformer-based methods.Transolver and Transolver++ use physics-aware attention, while PiT derives attention weights from pairwise spatial distances.Source: §2 · p. 3

Aerodynamic surrogates

• Aerodynamic surrogate work includes convolutional airfoil-field predictors and benchmarks AirfRANS and BlendedNet++ [17].AirfRANS provides two-dimensional airfoil flow fields, while BlendedNet++ provides three-dimensional blended-wing-body surface fields.Source: §2 · p. 3

Conditioning methods

• Global parameters may be concatenated with pointwise features or mapped by FiLM to channel-wise scale and shift coefficients.BlendedNet uses FiLM for pointwise surface-field prediction, and CoPiT combines multi-stage FiLM with PiT’s distance-based attention [13].Source: §2 · p. 3

3.1. Operator learning

• Operator learning approximates a solution operator from parameterized inputs to PDE solutions using a learned surrogate.For varying geometry, input and output domains and their function spaces may depend on the geometry.Source: §3.1 · p. 4
• Training uses input–output samples on finite meshes and minimizes an empirical loss evaluated on output points or a subsample.After training, the surrogate is designed to accept different input meshes and produce predictions at specified query locations.Source: §3.1 · p. 4

3.2. Position-induced attention

• Position-induced attention averages value-transformed features using weights determined by pairwise mesh distances rather than current feature content.A locality parameter can restrict keys to a nearest-neighbor fraction; its full value recovers a global kernel.Source: §3.2 · pp. 4–5
• Under the stated sampling and boundedness assumptions, discrete position-attention outputs converge in probability to those of a normalized Gaussian-type integral operator as sampling points increase.This is the discretization-consistency property of the attention mechanism.Source: §3.2 · p. 5
• PiT uses cross position-attention to map between input, latent, and output meshes, with self-attention blocks refining latent features.The architecture uses a latent mesh of moderate size.Source: §3.2 · p. 5

4.1. Overview

• CoPiT encodes input features onto a latent mesh, refines them with L FiLM-conditioned processor blocks, and decodes to the target mesh with a final FiLM stage.The same architectural skeleton is used for AirfRANS and BlendedNet++; encoder inputs, conditions, and auxiliary decoder channels are dataset-specific.Source: §4.1 · p. 5

4.2. Multi-stage FiLM conditioning

• FiLM maps the global condition vector to channel-wise scale and shift coefficients for instance-normalized features, with generators conditioning processor and decoder stages.The processor generator supplies L scale–shift pairs, while a separate generator supplies the decoder pair.Source: §4.2 · pp. 5–6, Figure 3 · p. 6

4.3. Processor and decoder with FiLM

• Each processor block modulates its concatenated attention features, while the decoder applies FiLM after cross-attention and any auxiliary-channel merge.With one attention head, the processor modulates 2D channels and the decoder modulates D channels in the reported models.Source: §4.3 · p. 7

4.4. Preservation of the positional kernel under FiLM

• For fixed conditions, bounded FiLM preserves convergence of an already convergent position-attention output, but this does not establish discretization consistency for the complete network.The complete composition also includes instance normalization, nonlinear multilayer perceptrons, latent-mesh selection, and auxiliary decoder features.Source: §4.4 · pp. 7–8

4.5. Separation of geometry and condition, and stability in the condition

• In FiLM-only CoPiT, encoded latent features and attention weights are condition-independent, and the prediction is Lipschitz continuous in the condition under the stated assumptions.A fixed-geometry sweep can reuse condition-independent computations; the Lipschitz bound depends on FiLM-generator and backbone constants and is acknowledged as pessimistic.Source: §4.5 · pp. 8–9

5.1.1. Datasets and metrics

• AirfRANS evaluates 2D RANS airfoil fields across full-data, scarce-data, Reynolds-number extrapolation, and angle-of-attack extrapolation settings.The full-data model uses geometry-only encoder inputs; the other listed settings concatenate freestream conditions while retaining FiLM conditioning.Source: §5.1.1 · p. 9
• Held-out field errors are scored in normalized space, and AirfRANS surface loss corresponds to wall-pressure MSE because wall velocity and turbulent viscosity are prescribed.BlendedNet++ test MSE averages normalized channel errors over surface points and cases, including the mean across three channels.Source: §5.1.1 · p. 10
• BlendedNet++ uses the official forward split of 8 992 / 1 000 / 2 498 train/validation/test cases and predicts three surface-flow quantities from six geometric channels and a 12-dimensional condition vector.The condition vector drives processor and decoder FiLM only in the main setup; a broadcast-condition hybrid is an ablation.Source: §5.1.1 · p. 10

5.1.2. Implementation

• AirfRANS models use a 7 000-point latent mesh and four processor blocks, while BlendedNet++ retains 10% of surface points by Z-order sampling and uses γ(c) = 1 + γraw(c).AirfRANS scores use the last-epoch checkpoint; BlendedNet++ evaluates the checkpoint with lowest validation loss once on the test set, with scores otherwise reported as single-run point estimates.Source: §5.1.2 · p. 10

5.1.3. Baselines

• AirfRANS comparisons include standard graph and transformer baselines plus matched PiT conditioning variants; BlendedNet++ comparisons include reimplemented baselines and concatenation-only and hybrid PiT variants.The BlendedNet++ baseline set includes FiLMNet, which uses FiLM without position-induced attention.Source: §5.1.3 · pp. 10–11

5.2.1. Full-data performance

• Full-field maps show recovery of the stagnation region, suction peak, pressure recovery, and near-wake, with residuals concentrated near the wall.The example visualizes spatial error structure rather than estimating dataset-level accuracy; reference and prediction share a color scale per variable, while error uses a separate scale.Source: §5.2.1 · p. 11, Figure 4 · p. 12
• CoPiT’s full-data mean relative errors for Cd and Cl are 0.4595 and 0.0467, respectively, so high drag rank correlation does not establish accurate absolute drag prediction.The reported drag ranking is relevant to design screening, but should not be read as an absolute-load accuracy result.Source: §5.2.1 · p. 11
• Approximately 54% and 24%: CoPiT reduces test surface and volume losses, respectively, relative to Transolver on the full-data split.The comparison is supported across three CoPiT seeds; each run improves on the Transolver result in all four reported metrics.Source: Table 1 · p. 11, §5.2.1 · p. 11

5.2.2. Scarce-data and extrapolation

• Under scarce data, CoPiT has the lowest surface loss and highest force-rank correlations, while Graph U-Net has the lowest volume loss.CoPiT’s comparatively large volume error is driven mainly by turbulent-viscosity prediction, so field accuracy is not uniformly superior.Source: Table 2 · p. 12, §5.2.2 · p. 12
• For Reynolds-number extrapolation, Transolver’s surface loss is approximately 1.2% lower, while CoPiT has the lowest volume loss and highest rank correlations.CoPiT’s mean relative drag error is approximately 221%, limiting the practical interpretation of its drag rank correlation.Source: Table 3 · p. 13, §5.2.2 · p. 13
• Under angle-of-attack extrapolation, Transolver is more accurate on both field losses and lift ranking, while CoPiT has the highest drag rank correlation.CoPiT’s mean relative errors of Cd and Cl are approximately 108% and 35%, respectively, so the drag-rank result is an ordering metric rather than evidence of accurate extrapolative loads.Source: Table 4 · p. 13, §5.2.2 · pp. 13–14

5.2.3. Conditioning ablation

• Matched full-data ablations show that FiLM generally outperforms condition input concatenation on surface loss, volume loss, and drag ranking, but the best FiLM placement depends on the metric.Decoder-only FiLM gives the lowest field losses in the family; complete CoPiT gives the highest drag rank correlation.Source: §5.2.3 · p. 14, Table 6 · p. 15

5.2.1. Full-data performance

• CoPiT achieves ρd ≈0.839 and ρl ≈0.999 on the 200-case full-data test set, versus Transolver’s ρd ≈0.145 and ρl ≈0.998.Lift ranking is strong for both, while drag ranking is substantially improved by CoPiT.Source: §5.2.1 · p. 11, Figure 7 · p. 15

5.3. BlendedNet++ results

• CoPiT attains the lowest overall normalized MSE (0.0518) and the lowest error on each of the three channels on the official forward test set.The single-run point estimate is under a common reimplementation protocol; scores use the per-channel z-score space used for training.Source: Table 7 · p. 16, §5.3 · p. 16
• For surface Cp on test_case_0674, relative L1 error is of order 0.06 for CoPiT versus 0.12 for Transolver, with weaker CoPiT error peaks near the wing–body junction.Both models recover the large-scale topology in this sample.Source: §5.3 · p. 16, Figure 8 · p. 16
• For streamwise skin friction C f,x on the same case, relative L1 error is of order 0.11 for CoPiT versus 0.35 for Transolver, with the largest residuals near the wing–body junction.Figure 9 shows CFD, CoPiT, and Transolver predictions and absolute errors on a shared color scale.Source: §5.3 · p. 17, Figure 9 · p. 17
• For spanwise skin friction C f,z on the same case, relative L1 error is of order 0.16 for CoPiT versus 0.46 for Transolver.The figures are qualitative and are not evidence of population-level superiority.Source: §5.3 · p. 17, Figure 10 · p. 18
• On the matched conditioning ablation, test MSE is 0.0741 with concatenation-only, 0.0570 with hybrid conditioning, and 0.0530 with FiLM-only CoPiT.The variants share a PiT backbone and Euclidean metric; these are single-seed point estimates on one random 20,000-point subset per case.Source: §5.3 · p. 18, Table 8 · p. 18

5.4. Computational efficiency

• Dense processor self-attention scales as O(LN^2_lttHD) in arithmetic operations and has quadratic attention storage unless evaluated in chunks.When attended keys per query are bounded, encoder and decoder costs scale linearly with full-mesh size; chunking can reduce peak memory without necessarily reducing arithmetic complexity.Source: §5.4 · pp. 18–19
• On AirfRANS, CoPiT is smaller than Transolver and trains in less wall-clock time, while their BlendedNet++ training times are comparable.The reported single-run training times use one NVIDIA A100; the table caption specifies 398 epochs for AirfRANS Full and 100 for BlendedNet++.Source: §5.4 · p. 19, Table 9 · p. 19
• 0.075 s with CoPiT versus 0.098 s with Transolver predicts a full AirfRANS mesh of about 1.8 × 10^5 points on the same GPU.Peak memory is 0.9 GB for CoPiT and 0.24 GB for Transolver; inference uses chunks of 32,000 points.Source: §5.4 · p. 19
• For multi-condition evaluation, computing geometry-dependent quantities once per airfoil (0.29 s) lets CoPiT evaluate 20 angles of attack in 0.013 s per condition, or 0.027 s including the geometric part.The result reproduces the standard forward pass exactly.Source: §5.4 · p. 19

5.5. Discussion

• High drag rank correlation does not by itself indicate accurate drag loads: the full-data run combines high ρd with approximately 46% mean relative drag error.The drag coefficient is a small residual between pressure and friction contributions, so relative error is sensitive to modest absolute deviations and small reference drag.Source: §5.5 · p. 19
• On AirfRANS, geometry-only PiT is not competitive, and FiLM variants improve surface loss and ρd relative to concatenation, but decoder-only FiLM can outperform complete CoPiT on field MSE.Complete CoPiT attains the highest ρd in the reported family, so the two FiLM stages are not uniformly additive across metrics.Source: §5.5 · p. 19
• On BlendedNet++, FiLM-only CoPiT outperforms both concatenation and hybrid broadcast-plus-FiLM on the same backbone and Euclidean metric.Increasing hidden width without FiLM does not close the AirfRANS gap between FiLM variants and concatenation.Source: §5.5 · p. 19

6. Conclusion

• CoPiT retains PiT’s position-induced Transformer backbone and uses multi-stage FiLM to modulate processor and decoder features according to freestream and design parameters.Pointwise modulation leaves position-attention weights unchanged and preserves convergence of an already convergent attention output under bounded modulation.Source: §6 · p. 19
• The convergence result for attention does not establish discretization consistency for the complete network.Additional analysis is required for the full composition.Source: §6 · p. 20
• On AirfRANS full-data tests, CoPiT achieves the lowest surface and volume losses among compared published baselines and higher drag rank correlation than Transolver.In scarce-data evaluation Graph U-Net has the lowest volume loss; Transolver is more accurate on angle-of-attack extrapolation fields and lift ranking, and high drag ranking does not guarantee accurate loads.Source: §6 · p. 20
• On BlendedNet++, CoPiT achieves the lowest overall normalized test-MSE point estimate among the reimplemented baselines.A matched ablation favors FiLM-only conditioning over input concatenation and hybrid broadcast-plus-FiLM.Source: §6 · p. 20
• Proximity-based attention may couple points close in ambient space but separated on the surface, AirfRANS errors remain concentrated near the wall, and training relies entirely on supervised data.Topology-aware metrics, residual-based regularizers, adaptive latent meshes, and tighter coupling to aerodynamic optimization are future-work directions.Source: §6 · p. 20

• Yongchen Fan contributed to conceptualization, methodology, software, investigation, validation, visualization, and both original-draft and review-and-editing writing.Source: CRediT authorship contribution statement · p. 21
• Junfeng Chen contributed to methodology, software, supervision, and writing review and editing.Source: CRediT authorship contribution statement · p. 21
• Zhonghua Qiao contributed to conceptualization, formal analysis, supervision, funding acquisition, and writing review and editing.Source: CRediT authorship contribution statement · p. 21
• Kailiang Wu contributed to methodology, formal analysis, funding acquisition, supervision, and writing review and editing.Source: CRediT authorship contribution statement · p. 21

Appendix A. Proof of Proposition 3

• The proof tracks feature differences in the RMS norm, using fixed point sets and row-wise Lipschitz maps to control them.Source: Appendix A · p. 21
• Instance normalization is bounded using its Jacobian, yielding an epsilon^-1/2 Lipschitz constant in the Euclidean norm and a corresponding RMS bound.Source: Appendix A · p. 21
• The conditional affine map is compared by decomposing differences into feature, scale, and bias terms, with the generators constrained by (23).Source: Appendix A · p. 21
• In each encoder block, the attention-output difference is bounded by the previous feature difference, after which the normalization and modulation lemmas are applied.Source: Appendix A · p. 21
• In the decoder, the auxiliary embedding is condition-independent, so the merge inputs differ only through attention output; inserting the derived bound proves (24)–(25).Source: Appendix A · p. 22