PaperlayerView exact revision on arXiv

Source-linked AI summary

VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows

Xingxin Yang, Zhan Zhang, Yichen Li, Juan Li

arXiv:2609.00507v1cs.LG

TL;DR

VATO addresses the difficulty of predicting unsteady separated-flow loads by coupling Vortex Force Map information to a geometry-aware neural operator. Across separated-aerofoil trajectories, its two variants improve flow prediction and aerodynamic force readouts, with different accuracy–inference-cost trade-offs.

  • Problem

    Unsteady separated-flow loads are difficult to predict efficiently because nonlinear separation and vortex shedding govern them, while uniform field-surrogate training does not identify load-relevant regions.

  • Method

    VATO couples the Vortex Force Map method to a geometry-aware transformer through training-only contribution-field supervision in VATO-S and force-relevant source prioritisation with residual cross attention in VATO-A.

  • Results

    VATO-S reduces velocity, pressure, and vorticity errors by 10.4%, 1.0%, and 15.6%, while VATO-A reduces them by 15.8%, 7.5%, and 31.2%; VATO-S has the lowest VFM-derived Drag error, whereas VATO-A has the lowest pressure-derived CL and CD errors.

  • Takeaways & Limitations

    VATO-S suits settings requiring unchanged inference cost, whereas VATO-A provides broader field and pressure-derived force accuracy at about 63% more measured inference time.

  • Takeaways & Limitations

    Evaluation covers unseen incidences on known geometries, uses one training seed per configuration, and quantifies bootstrap variation over geometry and incidence rather than retraining.

Abstract

from arXiv · show

Accurate prediction of unsteady separated flows is challenging because the aerodynamic loads depend on nonlinear separation and vortex-shedding dynamics. Although high-fidelity CFD resolves these mechanisms, its cost limits repeated use in design and control. Standard field-level surrogate training, however, does not distinguish the flow regions that contribute most strongly to the aerodynamic loads. We introduce VATO (Vortex-Force-Aware Transformer Operator), which couples the Vortex Force Map (VFM) method to a geometry-aware neural operator through two complementary mechanisms. VATO-S adds training-only supervision of the local VFM force-contribution field, with no increase in model size or inference cost. VATO-A uses VFM contribution and sensitivity fields to prioritise force-relevant source locations for residual cross attention. The methods are evaluated on unsteady CFD data for double-edged-plate aerofoils over 54 trajectories from nine geometries. Over lead times of 1-20~ms, VATO-S reduces velocity, pressure, and vorticity errors by 10.4\%, 1.0\%, and 15.6\%, respectively, while VATO-A achieves reductions of 15.8\%, 7.5\%, and 31.2\%. VATO-S gives the lowest VFM-derived drag error, whereas VATO-A gives the lowest pressure-derived lift and drag errors. Over lead times extending 50\% beyond the training range, VATO-A retains a 26.9\% reduction in vorticity error and larger improvements in all four force readouts, despite reduced gains in velocity and pressure. These results show that force-aware operator learning can improve both flow-field prediction and aerodynamic functional accuracy in unsteady separated flows.

1. Introduction

Unsteady separated-flow surrogates must capture nonlinear vortex dynamics while identifying the limited regions that determine aerodynamic loads. VATO couples VFM with a geometry-aware transformer through training supervision or force-relevant residual attention.

  • Motivation: High-fidelity CFD resolves separated-flow mechanisms but is too costly for repeated design and control simulations.Separated-flow loads depend on nonlinear, history-dependent separation, vortex formation, and shedding.
  • Motivation: Uniform field losses weight every mesh point equally, allowing wake and far-field regions to dominate despite loads originating in a small domain fraction.VFM provides a mechanics-based way to distinguish force-relevant regions.
  • Approach: VATO couples VFM to a geometry-aware transformer backbone through two interfaces.The approach uses a training-only contribution-field route and an architectural force-aware attention route.
  • Approach: VATO-S supervises the per-point VFM contribution field without changing architecture, parameter count, or inference cost.This variant is intended for deployments with fixed inference cost.
  • Approach: VATO-A prioritises Lift- and Drag-relevant regions for residual cross attention and provides the strongest field accuracy, especially for vorticity.It trades additional inference cost for broader field improvements.
  • Evaluation: The study evaluates held-out incidences and a 50% lead-time extension using independent pressure-surface and VFM volume force readouts.The attribution protocol separates vortex-force couplings from the training sampler.

2. Dataset

The dataset contains two-dimensional unsteady CFD trajectories for fourteen double-edged-plate geometries across thirteen angles of attack. Its split supports training, validation, held-out-incidence testing, and lead-time extrapolation, with CFD cross-checks against independent baselines.

  • Dataset scope: The dataset spans fourteen double-edged-plate geometries and thirteen angles of attack in two-dimensional unsteady CFD.The geometries vary through independent leading- and trailing-edge fold angles.
  • Geometry: The geometries range from a folded shape at θ1 = 156.2°, θ2 = 116.6° to the flat-plate reference θ1 = θ2 = 180°.A smaller fold angle denotes greater surface deflection.
  • CFD configuration: Each case uses an unstructured STAR-CCM+ mesh with 39,552–44,600 cells and 24,404–29,178 mesh points.The simulations use Martian atmospheric conditions and SST k–ω turbulence modeling.
  • CFD cross-check: The CFD setup agrees with transition-model and fully laminar reference results for time-averaged sectional lift across the operating envelope.The comparison supports geometrically fixed separation as the dominant setting at this Reynolds number.

3. Method

The method predicts future full-mesh flow fields from sampled current states using a geometry- and time-conditioned GAOT backbone. VATO variants couple VFM either through training loss or force-aware source prioritisation and residual attention.

  • Backbone: GAOT combines a MAGNO encoder, vision transformer, and MAGNO decoder to map unstructured physical coordinates through a regular latent grid.The encoder and decoder can operate on different point sets.
  • VATO variants: VATO-S retains the GAOT architecture and changes only the training objective through VFM contribution-field supervision.VATO-A instead couples VFM information through source prioritisation and residual attention.
  • Source prioritisation: VATO-A forms 256 source tokens, retaining 64 for geometric coverage and allocating 192 across signed force-contribution groups and VFM sensitivity.VFM values select locations but are not embedded in token content.
  • Training objective: The common objective combines physical-space velocity, pressure, and mesh-curl signed-log-vorticity losses.VATO-S additionally uses the training-only VFM contribution-field loss, whereas VATO-A does not.
  • Prediction task: The model predicts the future full-mesh field from the current sampled state, geometry condition, source time, lead time, and query coordinates.The training input uses a 12,000-point source set and the target is the future native CFD mesh field.

3.3. Pressure surface integral

Aerodynamic coefficients are recovered from predicted fields through a pressure integral on the body wall and a VFM volume integral in the wake and separation region. The VFM readout also produces local force-contribution maps that concentrate load-relevant information in a small subset of fluid points.

  • Pressure surface integral: The pressure readout integrates pressure over the aerofoil boundary and projects the resulting force onto lift and drag directions.The implementation evaluates the boundary integral by quadrature on native-mesh body edges.
  • VFM volume integral: The VFM volume integral provides a geometrically distinct lift and drag path supported on the wake and separation region rather than the body wall.It uses a geometry- and wind-axis-dependent auxiliary potential.
  • VFM volume integral: The VFM readout computes force from velocity and vorticity, with vorticity reconstructed from native-mesh velocity derivatives.Predicted-field readout applies the same mesh-curl operator to predicted velocity.
  • Validation: The VFM integral matches target force coefficients with correlations r = 0.9721 for CL and r = 0.9695 for CD on a representative trajectory.The pressure readout gives r = 1.0000 for CL and r = 0.9996 for CD on the same case.
  • Force concentration: The top 10% of fluid points contributes approximately 79% of absolute lift and 77% of absolute drag, while the top 20% contributes approximately 90% of both.These contribution maps motivate force-aware source selection.

3.5. Data preprocessing

The preprocessing converts native-mesh CFD fields into perturbation states by subtracting an inlet-background state, then standardises the residual and spatial-condition fields while preserving geometry and temporal inputs.

  • The learning state is the perturbation field obtained from the CFD field after subtracting the estimated inlet background.
  • Residual state and spatial-condition fields are standardised using training-set channel means and standard deviations.
  • Temporal scalars are scaled separately, aerofoil-interior points are masked, signed distance remains available as geometry conditioning, and vorticity is not predicted.

3.6. VFM contribution field supervision

VATO-S retains the GAOT architecture while adding training supervision for per-point VFM contribution fields whose sums produce discrete lift and drag coefficients.

  • VATO-S supervises two per-point contribution fields corresponding to the discrete VFM CL and CD.
  • The auxiliary objective compares predicted and target lift-and-drag contribution maps as one vector-valued field.
  • The predicted contribution field uses predicted velocity and curl-derived vorticity, while pressure does not enter the auxiliary equation.
  • Target contribution fields are training-only, so VATO-S uses the matched control’s inputs at inference without added parameters or inference-time operators.

3.7. Source prioritisation and residual cross attention

VATO-A prioritises force-relevant source locations using current vorticity, VFM contributions, and sensitivities, then injects ordinary source features through residual cross attention.

  • VATO-A combines current vorticity with VFM contribution and sensitivity fields to identify regions relevant to Lift and Drag.
  • The prioritisation uses RMS-normalised positive and negative contribution groups together with combined sensitivity-based groups and fallback allocation.
  • Exactly 64 tokens provide spatial coverage, while the remaining 192-token budget is allocated using detached input-frame prioritisation values.
  • Selected tokens carry ordinary state features and coordinates rather than the VFM values used for prioritisation.
  • Residual cross attention operates on full-frame source tokens and local neighbourhood tokens before decoding the predicted field.
  • The attention modules use multi-head scaled dot-product attention, with queries supplying the first argument and keys and values the second.

3.8. Training objective and protocol

Training combines physical-space field losses with flow-aware sampling, while VATO-S additionally optimises a VFM contribution loss; force-integral losses remain diagnostic readouts.

  • The shared physical-space objective uses squared relative L2 errors for velocity and pressure together with a mesh-curl vorticity term.
  • The displayed source-prioritisation example uses a trained 256-token budget and detached input-frame quantities.
  • VATO-S adds the VFM contribution loss with unit weight, whereas the reference, matched control, and VATO-A use only the shared objective.
  • Pressure-integral, scalar force-coefficient, and VFM-integral losses have zero weight and serve only as diagnostic readouts.
  • AdamW training uses 8,192 sampled pairs per epoch, batch size 8 per rank, four-rank distributed data parallelism, and 300 epochs.
  • Flow-aware trajectory sampling is shared by the matched control and both VATO variants, isolating coupling effects from sampler effects.

3.9. Evaluation metrics

The evaluation separates fluid-field errors from aerodynamic force errors and aggregates them across anchors, incidences, geometries, and lead-time windows. Field metrics exclude body-interior points, while force diagnostics apply pressure-surface and VFM volume operators to predicted and target fields.

  • Field errors are evaluated separately for velocity, pressure, and vorticity over the valid fluid-point set.
  • Both reference and predicted vorticity fields use the same native mesh-curl operator for reconstruction.
  • The four configurations differ in sampling, contribution-field loss, and architecture, with Table 2 reporting inference time on an NVIDIA H200.
  • Force diagnostics report sequence mean absolute errors for lift and drag using pressure-surface or VFM volume operators.
  • Errors are averaged hierarchically over anchors, incidences, and geometries, while lead-time-resolved figures retain the lead-time dimension before aggregation.

4. Results

Across the 54-trajectory benchmark, VATO-A provides the strongest overall field and pressure-derived force performance, while VATO-S is especially favorable for VFM-derived drag. Improvements remain visible beyond the training horizon and in held-out, separated-flow predictions, although VATO-A requires more parameters and inference time.

  • Training dynamics: 0.2325 validation field loss makes VATO-A the strongest configuration under the common 300-epoch protocol.This is a 34.9% reduction relative to the matched GAOT control.
  • Attribution and benchmark: Flow-aware sampling increases matched-GAOT velocity, pressure, and vorticity errors by 0.8%, 9.4%, and 0.7% over 1–20 ms relative to GAOT.It nevertheless reduces every force MAE, with gains ranging from 2.6% to 19.5%.
  • Direct-field benchmark: 15.8%, 7.5%, and 31.2% are VATO-A’s reductions in velocity, pressure, and vorticity error over 1–20 ms relative to GAOT.Over 21–30 ms, the corresponding reductions are 6.5%, 0.04%, and 26.9%.
  • Force diagnostics: VATO-A has the lowest pressure-derived lift and drag errors, while VATO-S has the lowest VFM-derived drag error.VATO-A reduces pressure-derived lift and drag MAE by 13.9% and 17.1% over 1–20 ms; VATO-S reduces VFM-derived drag MAE by 26.7%.
  • Efficiency: VATO-A uses 20.08 million parameters and 56.9 ms per sample, versus 19.09 million parameters and 34.8 ms for matched GAOT and VATO-S.Its additional cost comes from latent-patch, local-query, and output-query residual attention paths.
  • Lead-time behavior: 11.8% and 3.0% are VATO-A’s window-level equal-channel UVP improvements over GAOT in the trained and extrapolation windows.Pressure-derived force improvements remain roughly 12%–20% across the trained range, while beyond 20 ms they approach 20% for lift and 25% for drag near 25 ms.
  • Geometry and incidence: VATO-A is generally closer for the thinnest geometry, whereas VATO-S more closely follows the 12° response of the two thickest geometries.
  • Field predictions: Both VATO variants preserve vortex structures that GAOT loses at longer lead times, with VATO-A outperforming GAOT across all six test incidences.At 20 ms, VATO-A improves vorticity by 29%–43% and pressure by up to 71% at 9°.

5. Discussion and Limitations

VATO’s two interfaces improve different aspects of force-aware prediction, with VATO-A strongest for broad field and pressure-derived force accuracy. The evidence is bounded by coupled configuration effects and evaluation limited to known geometries and single training seeds.

  • Performance comparison: Both VATO configurations improve velocity, pressure, and vorticity over the matched control across both lead-time windows.
  • Functional accuracy: VATO-S gives the lowest VFM-derived Drag error, while VATO-A gives the lowest pressure-derived CL and CD errors.VATO-S adds no inference cost; VATO-A requires about 63% more measured inference time.
  • Extrapolation: Over a 50% temporal extension beyond training, VATO-A retains a 26.9% vorticity margin and increases all four force margins despite reduced pointwise gains.
  • Interpretation limits: VATO-A’s result reflects the complete configuration because prioritisation, residual attention paths, and trainable capacity change together.Isolating the prioritisation rule would require a geometry-only control matched in path and capacity.
  • Scope and limitations: The evaluation covers 54 trajectories from nine known geometries at unseen incidences, with one training seed per configuration and no unseen-shape or retraining assessment.Stronger claims would require multiple seeds, unseen geometries, cross-solver tests, calibrated force validation, and autoregressive evaluation.

6. Conclusion

VATO couples the Vortex Force Map method to a geometry-aware transformer at training and architectural interfaces. Across excluded incidences, VATO-A delivers the strongest field improvements, while the two interfaces span deployment choices from unchanged inference cost to additional inference time.

  • Conclusion: VATO couples the Vortex Force Map method to a geometry-aware transformer backbone through two interfaces and requires no additional flow solution.
  • Conclusion: VATO-S reduces velocity and vorticity error by 10.4% and 15.6%, while VATO-A reduces velocity, pressure, and vorticity error by 15.8%, 7.5%, and 31.2%.VATO-S has unchanged parameter count and inference cost; VATO-A uses 256 source locations and about 63% more measured inference time.
  • Deployment and limitations: The interfaces cover deployment conditions from training-only intervention with unchanged inference to additional inference time spent on the predicted field.Separating prioritisation effects from additional attention paths and testing unseen geometries remain next steps.
Loading 2609.00507v1…