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Shaping the Wind: Nested Potentials for Kinematically Admissible Urban Wind Prediction

Yidi Wang, Yunhe Zhang, Jiawei Gu, Ziyue Qiao, Pengyang Wang

arXiv:2610.07033v1cs.LG

TL;DR

Neural surrogates can make transient urban-wind prediction faster than repeated building-resolving LES, but accurate velocity prediction alone does not ensure mass conservation and wall impermeability. Sculpt parameterizes predictions with nested potentials to enforce both constraints, and across three evaluation settings it reduces 16-step mean squared error relative to nine baselines while keeping residuals near numerical precision.

  • Problem

    Neural surrogates offer a faster alternative to costly repeated LES, but geometry-conditioned velocity predictions are not restricted to satisfy local mass conservation and wall impermeability.

    Source: §1 · pp. 1–2

  • Method

    Sculpt uses nested volume and boundary potentials, with multiresolution volume potentials, to generate geometry-dependent admissible velocity updates.

    Source: §1 · p. 2, §3.2 · p. 6

  • Results

    Sculpt reduces 16-step mean squared error by 11.7–39.3% relative to nine baselines across three evaluation settings while maintaining constraint residuals near numerical precision.

    Source: §1 · p. 2

  • Takeaways & Limitations

    UrbanWindFlow provides 85 building-resolving LES sequences spanning morphologies and inflow conditions for joint evaluation of prediction accuracy and kinematic admissibility.

    Source: §1 · p. 2

    The guarantees rely on the regular staggered grid; extending them to unstructured meshes, cut cells, or curved manifolds requires a compatible discrete de Rham complex and boundary-edge assembly.

    Source: F LIMITATIONS AND FUTURE WORK · p. 21

Abstract

from arXiv · show

Predicting transient urban winds is fundamental to understanding urban microclimates and designing climate-resilient cities. Building-resolving large-eddy simulation produces detailed incompressible urban wind fields at substantial computational cost for each layout. Neural surrogates offer a faster alternative by learning to predict the evolution of velocity fields. However, minimizing velocity prediction error does not guarantee local mass conservation and wall impermeability, which together define kinematic admissibility. This limitation stems from an unconstrained output representation: geometry conditioning guides predictions but does not restrict them to admissible velocity fields. Correcting boundary violations in these outputs changes the flux balance in adjacent fluid cells and may consequently compromise local mass conservation. To address the challenge, we propose Sculpt, a nested potential framework that builds the coupled, geometry-dependent constraints directly into its parameterization. This nested parameterization generates divergence-free velocity updates through the discrete curl of a volume vector potential on the native three-dimensional staggered grid. A shared scalar potential constrains the vector potential's boundary values so that the same operator also enforces impermeability, without a per-step pressure projection. Because backpropagation through this curl attenuates large-scale gradient signals, we parameterize the volume potential at multiple resolutions to better capture large-scale flow structures. We introduce UrbanWindFlow, an LES dataset spanning urban morphologies and inflow conditions, to evaluate accuracy and kinematic admissibility together.

1 INTRODUCTION

Urban-wind surrogates can reduce the cost of repeated LES, but geometry conditioning alone does not ensure predictions satisfy local mass conservation and wall impermeability. Sculpt builds both constraints into a nested potential representation, and UrbanWindFlow supports joint evaluation of accuracy and admissibility across urban morphologies and inflow conditions.

1 INTRODUCTION

  • Geometry conditioning does not ensure kinematic admissibility: enforcing wall impermeability by masking can amplify divergence near building edges.Kinematic admissibility jointly requires local mass conservation and zero normal flow through solid surfaces.Source: §1 · pp. 1–2, Figure 1 · p. 2
  • Sculpt uses nested volume and boundary potentials on a staggered grid so the discrete-curl velocity updates satisfy conservation and impermeability without a per-step pressure projection.The volume potential yields divergence-free increments; boundary values derived from a shared scalar potential enforce zero normal flow.Source: §1 · p. 2
  • Multiresolution volume potentials give coarse flow structures dedicated prediction branches to address attenuation of large-scale gradients through curl backpropagation.The representation retains the same admissible readout.Source: §1 · p. 2
  • Across three evaluation settings, Sculpt reduces 16-step mean squared error by 11.7–39.3% relative to nine baselines while keeping constraint residuals near numerical precision.The settings cover unseen layout instances, unseen morphology classes, and held-out inflow combinations.Source: §1 · p. 2
  • UrbanWindFlow provides building-resolving LES sequences spanning urban morphologies and inflow conditions for evaluating accuracy, admissibility, and transfer.The dataset contains 85 sequences.Source: §1 · pp. 2–3

2 PRELIMINARIES

The preliminaries specify a staggered-grid representation in which layout determines fluid cells and boundary faces. They define kinematic admissibility as joint local mass conservation and wall impermeability, then formulate history-based prediction with admissibility required throughout rollout and across geometry and inflow shifts.

2 PRELIMINARIES

  • Urban-wind fields are discretized on a uniform staggered grid, with horizontal periodicity and ground and lid boundaries; the layout mask marks fluid and solid cells and determines velocity faces.UrbanWindFlow contains 85 building-resolving LES sequences of 100 snapshots each, spanning morphologies and inflow conditions; snapshots satisfy both kinematic conditions within numerical precision.Source: §2 · p. 3, §2.1 · p. 3

2.1 KINEMATICALLY ADMISSIBLE URBAN WIND FIELD

  • Kinematic admissibility requires zero finite-volume divergence in every fluid cell and zero normal velocity on impermeable faces.The admissible space is the geometry-specific kernel of the joint constraint operator; the two conditions share boundary-face velocities, so changing one can affect the other.Source: §2.1 · p. 3

2.2 PROBLEM FORMULATION

  • The prediction task advances recent velocity snapshots autoregressively using layout and inflow, with admissibility required at every rollout horizon and evaluation across three geometry and inflow shifts.Rather than penalizing violations or applying a per-step pressure projection, the structured representation uses a discrete curl of a volume potential and boundary values assembled from a shared scalar potential.Source: §2.2 · p. 4

3 METHODOLOGY

Sculpt predicts velocity updates by assembling volume and boundary potentials, then applying a discrete curl that enforces mass conservation and wall impermeability. Its multiresolution volume readout addresses the curl’s wavelength-dependent gradient reweighting, while a surface graph network predicts geometry-dependent boundary potentials.

3.1 NESTED VOLUME-BOUNDARY POTENTIAL FRAMEWORK

  • The assembled potential’s discrete curl produces divergence-free velocity updates, while shared scalar-potential differences set boundary values so circulation around boundary faces vanishes.The assembly retains predicted interior-edge values and replaces boundary-edge values; the resulting updates satisfy local mass conservation and wall impermeability.Source: §3.1 · p. 5

3.2 MULTIRESOLUTION VOLUME POTENTIAL

  • Because backpropagation through the curl attenuates long-wavelength gradients and amplifies short-wavelength ones, each U-Net resolution contributes a volume-potential candidate.The multiresolution readout raises the gradient share carried by large eddies and lowers the share carried by small ones.Source: §3.2 · p. 6, Figure 4 · p. 6

3.3 SURFACE GRAPH NETWORK FOR THE BOUNDARY POTENTIAL

  • A surface graph network predicts the boundary scalar potential by aggregating along boundary edges, matching the connectivity of the surface mesh.This avoids treating nearby façades across a narrow street as surface neighbors merely because they are close in Euclidean distance.Source: §3.3 · p. 6
  • Two harmonic coefficients represent periodic global circulations that the boundary scalar-potential gradient cannot express, and they control the mean horizontal wind.The coefficients are taken from the lid means of the horizontal components emitted by the volume module.Source: §3.3 · p. 7

4 EXPERIMENTS

The experiments evaluate wind-field prediction across unseen layouts, morphologies, and inflows, measuring both velocity error and kinematic admissibility. Sculpt leads the reported comparisons, while ablations and visualizations examine its components and remaining errors.

4.1 EXPERIMENT SETUP

  • Testing separates unseen layouts, unseen morphologies, and held-out inflow combinations; each test sequence rolls out from the same 18 start frames.The reported metrics are fluid-cell velocity MSE across rollout horizons and worst-case divergence and wall-normal velocity over test sequences and horizons.Source: §4 · pp. 7–8, §4.1 · p. 8

4.2 EXPERIMENTAL RESULTS

  • Sculpt lowers k=16 error by 29% versus the strongest baseline averaged across three settings, and by 61.9% at k=1; it still leads unseen inflows by 11.7%.The unseen-inflows setting is hardest for learned models, and POD overtakes them at k=16.Source: §4.2 · pp. 8–9
  • AFNO reaches an MSE of 1.6×10^3 and divergence of 5.9×10^2 under inflow shift, while U-Net's worst face permits flow at 1.06, 1.05, and 1.30 times free-stream speed.U-Net's boundary penetration is more than five orders of magnitude above Sculpt.Source: §4.2 · p. 9
  • Removing the null-space map or using a soft penalty increases error and constraint violations; post-projection restores constraints but requires a linear solve each step and trails Sculpt.Removing the shared boundary potential preserves the divergence floor but permits wall flow up to 1.72 times free-stream speed and raises 16-step error by 19.3–39.7%.Source: §4.2 · p. 9
  • At 16 steps, Sculpt's error occupies a smaller volume; its mean local error falls from 9.3% next to walls to 3.3% beyond 16 m, while the least accurate shared region is the roof-level shear layer.The region where all four methods exceed 12% of free-stream speed averages 21.5 m in height, between the mean roof height of 12.9 m and the tallest building at 36 m.Source: §4.2 · p. 9, Figure 5 · p. 9

5 CONCLUSION

The conclusion presents Sculpt as a nested potential approach to geometry-compatible urban wind prediction and introduces UrbanWindFlow for joint evaluation of accuracy and admissibility. It also points readers to code, implementation details, and publicly available datasets.

5 CONCLUSION

  • Sculpt uses nested volume and boundary potentials to produce divergence-free updates while enforcing wall impermeability, with the guarantee built into the parameterization.Across three UrbanWindFlow settings, it lowers MSE by 11.7–39.3% against nine baselines while keeping both residuals below 10^-6.Source: §5 · p. 10
  • UrbanWindFlow contains building-resolving LES sequences and three evaluation settings for assessing accuracy and kinematic admissibility across morphologies and inflow conditions.The settings address unseen layout instances, unseen morphology classes, and held-out inflow combinations.Source: §5 · p. 10
  • The authors provide an anonymous source-code link, model and training details in Appendix C.1, and public datasets with statistics summarized in Section 4.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 PROOFS OF PROPOSITIONS 3.1 AND 3.2

The proofs establish that the discrete curl yields divergence-free updates and zero boundary-face updates under the nested boundary construction. The appendix also explains how curl readout attenuates large-scale gradients, motivating multiresolution parameterization.

A.1 PROOFS OF PROPOSITIONS 3.1 AND 3.2

  • Summing the curl update over a cell cancels every shared edge contribution in opposite incidence pairs, giving zero discrete divergence for any potential and mask.Source: §A.1 · p. 15
  • Boundary-face updates vanish because the discrete gradient term has zero circulation and the closed boundary 1-cochain also has zero circulation around each boundary face.Source: §A.1 · p. 15

A.2 SPECTRAL GAIN OF THE CURL READOUT

  • Curl readout attenuates potential gradients at large scales by a factor of order |ξ|h relative to grid-scale modes; the shared scalar potential's gradient-then-curl path squares that attenuation.The boundary argument is exact on flat portions and serves as a guide elsewhere; this scaling motivates multiresolution readout.Source: §A.2 · p. 16

B.1 DATASET GENERATION

UrbanWindFlow is generated from building-height layouts and PALM LES runs across varied morphologies and inflow conditions. Its documented grid, boundary setup, and inflow combinations define the dataset's simulation scope.

B.1 DATASET GENERATION

  • Layouts are filled building columns tiled from sixteen blocks, with morphology and random seed determining block types, footprints, heights, and street-grid offsets.Street widths are 24, 16, and 6 m on a 128 m block rhythm; storeys are 3 m high and building heights are quantized to the 2 m grid.Source: §B.1 · p. 16
  • PALM LES runs use a 24×128×128 grid at 2 m spacing, horizontally periodic boundaries, rough ground, and a free-slip lid; the solver blocks building cells and uses the native Arakawa-C staggered grid.Source: §B.1 · pp. 16–17
  • The 85 sequences span three wind directions, three free-stream speeds, and convective, neutral, or stable conditions; three combinations are held out for testing.Each run integrates for 1,000 s before 100 consecutive one-second snapshots are collected.Source: §B.1 · p. 17

B.2 COMPARISON WITH EXISTING BENCHMARKS

UrbanWindFlow addresses a benchmark gap for evaluating transient wind prediction over resolved urban geometry. Its evaluation and training setup separates layout, morphology, and inflow generalization while keeping methods on common inputs and splits.

B.2 COMPARISON WITH EXISTING BENCHMARKS

  • No publicly released dataset provides instantaneous volumetric velocity fields over resolved urban geometry, motivating UrbanWindFlow’s solid-occupancy masks and procedurally varied layouts.The benchmark reference is Gen4Turbulence’s tensor layout, resolution class, and snapshot count.Source: §B.2 · p. 17

C.1 IMPLEMENTATION DETAILS.

  • The evaluation uses sequence-separated tests for unseen layouts, unseen morphologies, and held-out inflows, with test rollouts starting at 18 frames per sequence.Training and validation frames have a six-frame gap; test sequences are absent from training at every frame.Source: §C.1 · pp. 17–18
  • All methods use four history frames, identical inputs and splits, and 120 training epochs; Sculpt uses reconstruction loss alone and pushforward training with probability p.With probability p, the most recent input is replaced by a gradient-free one-step prediction and the target advances accordingly.Source: §C.1 · p. 18

C.2 ABLATION VARIANTS

The ablations distinguish constraint-enforcing output choices from other design choices, while the unseen-inflows evaluation compares Sculpt with neural and reduced-order baselines. Sculpt leads the neural baselines across rollout horizons in that setting.

C.2 ABLATION VARIANTS

  • The constraint-focused variants remove the null-space representation, add soft divergence and wall penalties, apply post-prediction pressure projection, or omit boundary-potential substitution.The post-projection variant clears solid-face normal velocity and solves a Poisson problem at each predicted step.Source: §C.2 · p. 18
  • Other variants predict absolute fields instead of increments, remove multiscale potential resolutions, or replace learned boundary fusion with uniform averaging.These choices are described as not required for either kinematic guarantee.Source: §C.2 · p. 18

D.1 PERFORMANCE ON UNSEEN INFLOWS SETTING

  • At k=1 and k=16, Sculpt is ahead of the strongest neural baseline by 55.7% and 33.2%, respectively, on unseen inflows.At k=16, POD overtakes every neural baseline, and AFNO falls outside the table’s range.Source: §D.1 · p. 18, Table 7 · p. 19

D.2 ABLATION ON UNSEEN INFLOWS SETTING

On unseen inflows, Sculpt remains first at both ablation horizons, with multiresolution potential and learned fusion showing larger reported costs than on unseen layouts. The model’s fixed-width complexity is linear in grid size and comparable to convolutional baselines.

D.2 ABLATION ON UNSEEN INFLOWS SETTING

  • On unseen inflows, removing multiresolution potential or learned fusion costs 30.7% or 51.6% at k=16, versus 1.8% or 16.0% on unseen layouts.Sculpt ranks first at both reported horizons in the unseen-inflows ablation.Source: §D.2 · p. 19, Table 8 · p. 19

D.3 COMPUTATIONAL COMPLEXITY

  • Sculpt’s stated complexity is O(C0^2N + KcB^2|N| + N), reducing to O(N) at fixed widths and comparable to convolutional baselines.The boundary complex spans 9.6%–12.3% of cells; assembly and discrete curl require O(|E∂(g)| + N) work without parameters or a linear system.Source: §D.3 · p. 20

D.4 PARAMETERS ANALYSIS

The parameter analysis selects a shared message-passing depth and pushforward probability across evaluation settings, and the visualizations examine how rollout errors evolve. The related-work discussion distinguishes transient neural PDE surrogates from other urban-wind and constraint-enforcement approaches.

D.4 PARAMETERS ANALYSIS

  • Across all three settings, 16-step MSE rises when pushforward probability reaches 0.9; K is relatively stable, so the evaluations use K=4 and p=0.5.The authors attribute the high-p increase to fewer steps anchored to ground truth and resulting training instability.Source: §D.4 · p. 20, Figure 6 · p. 20

D.5 VISUALIZATION

  • Across three test sequences, error grows with horizon; baseline error shells merge into connected sheets by k=8–16 while Sculpt’s remain separated.The regions that fill first are above roofs, where Section 4.2 locates every method’s largest error.Source: §D.5 · p. 20

E RELATED WORKS

  • Urban-wind surrogates commonly return steady or time-averaged, often two-dimensional fields; neural PDE surrogates instead roll out transient three-dimensional fields, with layout as input (Gupta & Brandstetter, 2022; Brandstetter et al., 2022).Sculpt additionally uses layout to determine the space in which its output is expressed.Source: E RELATED WORKS · pp. 20–21
  • Penalty, projection, and structured-output methods enforce constraints through training residuals, post-inference correction, or construction; immersed building surfaces couple the two constraints on shared degrees of freedom.The adjoint discrete curl rescales gradients by wavenumber, which the multiresolution read-out addresses.Source: E RELATED WORKS · p. 21

F LIMITATIONS AND FUTURE WORK

Sculpt’s guarantees rely on a regular staggered grid, and extending the boundary construction to other discretizations remains open. All four surrogates also struggle in the roof-level shear layer, where squared-error training underrepresents the most fluctuating flow.

F LIMITATIONS AND FUTURE WORK

  • Sculpt’s construction is tied to the incidence structure of a uniform staggered grid.Source: F LIMITATIONS AND FUTURE WORK · p. 21
  • Extending the construction to unstructured meshes, cut cells, or curved manifolds requires a compatible discrete de Rham complex and boundary assembly; the latter remains open.Source: F LIMITATIONS AND FUTURE WORK · p. 21
  • All four learned methods are least accurate in the least steady roof-level shear layer, where squared-error training underrepresents the most fluctuating flow component.A one-step squared-error map approaches the conditional mean of the next frame; whether probabilistic or ensemble objectives recover that component, and at what pointwise-error cost, remains future work.Source: F LIMITATIONS AND FUTURE WORK · p. 21
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