Fourier neural operators for real-time prediction of urban airflow and pollutant dispersion

Abstract

Building-resolving computational fluid dynamics can predict urban wind and pollutant dispersion with useful physical fidelity, but the simulation cost is too high for rapid emergency response, design iteration, or real-time exposure screening. We present a Fourier neural operator surrogate for steady and quasi-steady urban airflow and passive-scalar dispersion over complex building arrays. The model maps rasterised building height, signed-distance geometry, inflow wind profile, thermal-stability class, surface roughness, and source terms to three-dimensional mean velocity, turbulence-intensity proxy, and concentration fields on a 256 x 256 x 48 grid. Training data comprise 8,400 building-resolving simulations over a fictional 3.2 km x 3.2 km district, with wind directions, speeds, atmospheric stability, and source locations sampled by Latin hypercube design. The best model used a geometry-aware Fourier neural operator with local convolutional lifting and spectral mode truncation varying by height. On withheld geometries and meteorological conditions, it achieved 7.8% normalised mean absolute error for horizontal wind speed, 11.6% for vertical velocity, and 13.9% for concentration above a 1% source-normalised threshold. Inference required 0.21 s on a single GPU and 3.8 s on a workstation CPU, compared with 2.4-7.1 h for the reference solver. The surrogate preserved plume centroids, street-canyon recirculation zones, and roof-level acceleration in most cases, but under-predicted intermittent high-concentration pockets in weak-wind stable conditions. The results show that neural operators can support rapid urban dispersion screening when trained on physically diverse simulations and constrained by mass-balance diagnostics, but they should not replace high-fidelity CFD for final regulatory assessment.

Introduction

Urban pollutant dispersion depends on building geometry, inflow profile, atmospheric stability, source location, heat release, and street-scale recirculation. Computational fluid dynamics is widely used because it can represent building-scale flow features that Gaussian plume models cannot, but it is expensive and sensitive to mesh, boundary conditions, turbulence closure, and validation protocol [5,6,7,8,9]. Large-eddy simulation is especially informative for unsteady street-canyon dispersion, but it is rarely fast enough for operational scenario exploration [10,11,12,13,14].

Neural operators offer a different kind of surrogate. Instead of learning a mapping between fixed-dimensional parameter vectors and scalar outputs, they learn mappings between functions, such as boundary conditions, geometry fields, and solution fields. Fourier neural operators use spectral convolutions to represent long-range interactions efficiently and have become a standard reference point for parametric partial differential equations [1,2]. Related operator-learning approaches, including DeepONet and physics-informed neural operators, show that learned solution operators can generalise across boundary conditions and source terms when the training distribution is well designed [3,4].

Urban airflow is a demanding test for this idea. The domain is geometrically rough, the solution contains sharp shear layers and wakes, and pollutant concentrations can be dominated by rare local trapping events. Recent urban-flow surrogate studies and uncertainty-aware dispersion models suggest that machine learning can accelerate prediction, but also that validation must be more stringent than visual agreement [15,16,17].

This article reports a fictional but technically plausible Fourier neural operator workflow for rapid prediction of urban airflow and passive-scalar dispersion. The model is trained only on synthetic CFD data, not on proprietary sensor records or unpublished field measurements. The aim is to quantify speed, accuracy, failure modes, and diagnostics needed before such a surrogate could be used in planning or emergency support.

Reference CFD dataset

The reference dataset was generated for a fictional 3.2 km x 3.2 km urban district with a mix of street canyons, mid-rise blocks, open plazas, and three high-rise clusters. Building footprints were procedurally generated from statistical distributions of block size, height, spacing, and road width, then manually screened to remove impossible overlaps. This avoids copying a real city while preserving the geometric complexity needed for urban-flow learning.

Each CFD case used a building-resolving finite-volume solver with wall-modelled large-eddy simulation for 1,600 high-fidelity cases and steady Reynolds-averaged simulations for 6,800 lower-cost cases. The RANS cases expanded the parameter space, while the LES cases anchored unsteady recirculation and plume-spread statistics. The training target for the neural operator was the time-averaged LES field where available and a bias-corrected RANS field elsewhere.

Wind directions were sampled every 10 deg with random perturbations, inflow speeds ranged from 1.5 to 12 m s-1 at reference height, and atmospheric stability was represented by five Monin-Obukhov classes. Passive sources included ground-level road emissions, point releases at 2-20 m height, and roof exhausts. Source strengths were normalised so the model learned dispersion response rather than emission inventory.

The computational grid was resampled to 256 x 256 horizontal cells and 48 stretched vertical levels up to 320 m. The learning problem therefore used a uniform analysis grid, even though the CFD solver used body-fitted and refined near-wall meshes. This resampling is a limitation, but it is also what makes city-scale operator learning practical.

Input and output representation

The input tensor contained building height, ground mask, signed distance to the nearest wall, height-normalised roughness, inflow speed profile, wind direction encoded as sine and cosine, stability class, source height, source mask, and a scalar source-intensity channel. For stability, a continuous bulk Richardson proxy was also included, because categorical stability labels caused discontinuities near class boundaries.

The output tensor contained mean velocity components u, v, and w, a turbulence-intensity proxy derived from resolved kinetic energy where LES was available, and source-normalised concentration. Concentration was represented in log1p form to avoid allowing high-concentration cells near the source to dominate the loss. During evaluation, errors are reported both in log concentration and in physical source-normalised concentration.

Geometry encoding was critical. A building-height raster alone produced plausible roof-level flow but poor street-canyon recirculation. Adding signed distance and wall-normal channels reduced near-wall velocity error by 18% and concentration error by 11%. This is consistent with the basic CFD lesson that geometry and boundary treatment, not only free-stream wind, control near-field dispersion [5,8].

The output was masked inside buildings. Loss terms were computed over fluid cells, with additional penalties for no-flow-through-wall violations at building boundaries and for domain-integrated passive-scalar imbalance.

Fourier neural operator architecture

The model used a four-level Fourier neural operator. A local convolutional lifting network first embedded the input fields into 64 channels. Each operator block combined spectral convolution in horizontal planes, depthwise vertical mixing, and a local residual convolution. The number of retained Fourier modes decreased with height: more modes near the canopy, fewer in the smoother upper boundary layer. This reduced memory while preserving street-scale detail.

A pure FNO trained on the full 3D grid was accurate but too memory intensive for rapid iteration. The final model used separable horizontal spectral transforms and learned vertical coupling. This is not a new neural-operator theory contribution; it is an engineering adaptation for a tall, anisotropic urban domain.

The loss function combined velocity L1 loss, concentration log-L1 loss, gradient loss near building edges, source-integral conservation, and a weak divergence penalty on the mean flow. The divergence penalty was not strong enough to enforce incompressibility exactly, but it reduced unphysical mass imbalance in narrow streets. Physics-informed neural-operator ideas motivated these diagnostic penalties, although the model was trained primarily from CFD data rather than from residual collocation [4].

Training used 7,200 cases, with 600 validation cases and 600 test cases. A second test set contained 180 out-of-distribution cases with unseen high-rise clusters and weak-wind stable inflow. This split was chosen because random case splits can overstate surrogate generalisation when geometries are similar.

Training protocol

The network was trained for 180 epochs using AdamW with cosine learning-rate decay. The first 40 epochs used velocity-only loss so the model learned the mean flow before concentration. Concentration and conservation penalties were then ramped in over 30 epochs. This staged training avoided early collapse to trivial low-concentration predictions in cases where plumes occupied a small fraction of the domain.

Training data were augmented by horizontal reflection, small wind-direction perturbation, and source-strength rescaling. Rotation augmentation was not used because the urban grid and prevailing-street statistics were not rotation invariant. Meteorological channels were normalised by physically meaningful reference values rather than by global dataset mean alone.

The model ensemble used five random initialisations and two data-order seeds. The production prediction is the ensemble mean, while ensemble spread is used as a crude epistemic uncertainty indicator. This is cheaper than a full Bayesian operator model but provides useful flags for unusual geometries and stable weak-wind cases.

The final checkpoint had 46 million trainable parameters. Training required 42 h on four A100-class GPUs. Inference for one 3D case required 0.21 s on one GPU and 3.8 s on a 24-core workstation CPU, including preprocessing and postprocessing.

Accuracy on withheld cases

On the standard withheld test set, normalised mean absolute error was 7.8% for horizontal wind speed, 11.6% for vertical velocity, and 13.9% for concentration in cells exceeding 1% of the source-normalised maximum. Plume-centroid horizontal error was 8.6 m median and 27.4 m at the 95th percentile. Peak-concentration error was larger, with median under-prediction of 16%, because the model smoothed intermittent high-concentration pockets.

Street-canyon recirculation was captured when the canyon aspect ratio and wind angle were represented in the training set. For unseen high-rise clusters, the model predicted roof-level acceleration accurately but under-estimated downwash behind isolated towers. This failure mode is familiar from CFD practice: separated flow around isolated tall buildings is sensitive to turbulence modelling and grid resolution [7,18].

Concentration exceedance classification was evaluated for thresholds relevant to screening rather than for exact concentration values. The model achieved area-under-curve 0.92 for identifying cells above 10% of the case maximum and 0.84 for cells above 30%. False negatives clustered in low-speed stable cases, where plume meandering and trapping produce small high-concentration pockets.

The speedup relative to reference simulation was 2,800-120,000 times depending on whether the baseline was RANS or LES and on mesh size. The number is impressive but should not be over-read. The expensive part is generating the training dataset; the surrogate is fast only after that investment is made.

Comparison with baseline surrogates

We compared the FNO with a U-Net, a graph neural network on building cells, and a reduced-order proper-orthogonal-decomposition surrogate. The U-Net performed well for concentration near the source but lost long-range plume steering. The graph model handled irregular geometry but was slower and less accurate on roof-level acceleration. The reduced-order model was fast but failed when source location changed.

The FNO advantage was largest for wind-direction and source-location interpolation. Spectral convolution gave the model a way to represent domain-scale pressure adjustment and plume advection without stacking many local convolution layers. This is the same operator-learning motivation that made FNO attractive for parametric PDEs [1,2]. Other neural-operator variants may improve local feature recovery, but the comparison here keeps the architecture deliberately close to the established Fourier formulation [19].

The uncertainty-aware surrogate model from recent urban dispersion work provides an important comparison point [16]. Our ensemble spread correlated with error, but it was under-dispersed for out-of-distribution geometries. We therefore calibrate uncertainty with validation residuals and recommend treating high ensemble spread as a warning flag, not as a full probabilistic forecast.

The FNO also compared favourably with Seveso-type dispersion surrogate modelling in speed, but the problem setting differs [15]. Industrial accidental-release models often operate over smoother terrain and coarser grids. Urban canopy flow requires much more geometric detail.

Real-time scenario workflow

The operational workflow starts with a building-height raster, an inflow profile from a weather model or mast, and one or more emission-source masks. The model then generates velocity and concentration fields for all wind-direction branches in a few seconds. A simple postprocessor computes exposure zones, plume centroids, building-facade exceedance maps, and uncertainty flags.

For emergency use, we do not recommend choosing a single deterministic wind direction. Instead, the system samples wind direction and stability from a short-range meteorological ensemble and produces exceedance probability maps. The neural operator makes this feasible because hundreds of scenario evaluations cost less than one conventional CFD run.

For design use, the workflow supports rapid comparison of building layouts, street-ventilation changes, and source-control strategies. However, geometry changes outside the training distribution must be flagged. A new high-rise podium or enclosed atrium may require additional CFD cases and fine-tuning before predictions are credible.

The model is therefore best understood as a screening layer. It helps decide which scenarios deserve high-fidelity simulation or field measurement. It does not remove the need for CFD guidelines, validation, or expert review [6,7,8].

Physical diagnostics

We used three diagnostics to detect unphysical predictions. First, domain-integrated passive-scalar imbalance checks whether the emitted mass approximately leaves the domain or remains stored in the predicted field. Second, wall-normal flow leakage checks no-flow-through-wall consistency on building masks. Third, kinetic-energy spectra compare predicted and reference canopy-scale energy distribution.

The raw FNO sometimes produced weak wall leakage at sharp corners. Adding the signed-distance channel and boundary penalty reduced the median leakage by 63%. Remaining leakage was concentrated at one-cell diagonal building contacts, which are partly a rasterisation artefact. This is why geometry preprocessing matters as much as network architecture.

The model preserved the k^-5/3-like inertial-range slope poorly because it predicts mean fields and a turbulence proxy rather than instantaneous LES snapshots. This is acceptable for screening mean exposure, but it means the surrogate cannot reproduce gust statistics or peak short-term exposure without a separate fluctuation model.

Mass-balance diagnostics caught several cases where visual plume shape looked plausible but concentration magnitude was wrong. We include these diagnostics in the article because surrogate modelling should not be judged only by attractive contour plots.

Out-of-distribution behaviour

The out-of-distribution test set exposed the limits of the surrogate. Unseen tower clusters produced local vertical-velocity errors above 30%. Weak-wind stable cases produced the largest concentration under-prediction, especially in courtyards connected to narrow streets. These are the same conditions where conventional modelling is difficult and where field validation would be most valuable.

The ensemble uncertainty was helpful but incomplete. It flagged 71% of cases whose concentration error exceeded 25%, but missed several stable low-wind cases where all ensemble members made the same smooth-plume error. A model can be confidently wrong when the training data omit a physical regime.

Fine-tuning with 80 additional CFD cases from the OOD geometries reduced concentration error by one third. This suggests a practical adaptive workflow: deploy the base FNO for screening, identify flagged geometry/weather regimes, run targeted CFD, and update the surrogate. The cost is still far lower than building a new CFD database from scratch.

We deliberately avoid claiming extrapolation to arbitrary cities. The learned operator generalises within a family of urban morphologies and meteorological ranges. It does not infer fluid mechanics from first principles in a completely new regime.

Discussion

The main result is that Fourier neural operators can provide real-time, three-dimensional urban airflow and dispersion predictions at useful screening accuracy when trained on a diverse CFD ensemble. The model is fast enough for scenario ensembles, source-location sweeps, and layout comparisons. Its accuracy is strongest for mean wind and plume footprint, weaker for vertical velocity and local concentration peaks.

This is not a replacement for building-resolving CFD. The surrogate inherits the assumptions of its training simulations: turbulence closure, boundary conditions, neutral or simplified stability, wall functions, and source treatment. Reviews of urban CFD repeatedly warn that boundary conditions and validation protocol can dominate results [5,6,7,8,9]. A neural operator trained on biased CFD will reproduce that bias quickly.

The most promising role is interactive modelling. Planners and emergency analysts can explore hundreds of plausible scenarios, then send the most consequential or uncertain cases to high-fidelity simulation. This kind of human-in-the-loop workflow is more realistic than expecting a single neural network to become a regulatory solver.

The approach also suggests new data priorities. Additional CFD cases are most valuable where the surrogate is uncertain or wrong: stable weak winds, isolated towers, deep courtyards, thermal plumes, and unusual source heights. Active learning could reduce the number of expensive simulations needed to improve coverage.

Limitations

The first limitation is the synthetic city. It contains realistic geometry statistics but does not include vegetation, traffic heat, moving vehicles, detailed facade roughness, or real terrain. Applying the model to a real city would require geometry cleaning and additional CFD cases.

The second limitation is meteorology. Stability is represented by a bulk scalar rather than by a fully coupled surface-energy-balance model. Convective and strongly stable boundary layers are therefore simplified. The model should not be used for heat-wave ventilation or nocturnal inversion studies without additional training data.

The third limitation is concentration physics. The pollutant is passive and non-reactive. Chemistry, aerosol deposition, buoyant fire plumes, and dense-gas effects are outside scope. Some accidental-release scenarios would require different governing equations and therefore a different operator.

Finally, the training data are expensive. Neural operators shift cost from online prediction to offline dataset generation. That trade can be worthwhile for repeated scenario evaluation, but not for one-off studies where only one high-fidelity CFD case is needed.

Conclusion

A geometry-aware Fourier neural operator trained on building-resolving CFD can predict urban mean airflow and passive-scalar dispersion in seconds, with errors low enough for rapid screening and scenario triage. The model captured most street-canyon recirculation, roof-level acceleration, plume steering, and source-location effects in withheld cases, while revealing predictable weaknesses in weak-wind stable conditions and unseen high-rise geometries.

The practical lesson is disciplined acceleration. Neural operators should be paired with CFD-quality training data, physical diagnostics, uncertainty flags, and targeted high-fidelity follow-up. Used that way, they can make urban dispersion modelling more interactive without pretending that fast inference is the same as validated physics.

Data and code availability

The supplementary archive contains the synthetic geometry generator, CFD case metadata, processed training tensors, model configuration files, trained checkpoints, evaluation scripts, withheld-case predictions, and diagnostic notebooks. Full raw LES fields are provided for a representative subset; the complete raw CFD archive is listed by checksum because of its size.