Fourier neural operator for real-time simulation of 3D dynamic urban microclimate

1 Introduction

• Urban features shape local living environments and give rise to microclimates where most human activities take place.Source: §1 · p. 2
• RANS models are more practical for urban wind prediction than LES because they run faster, although they may be less accurate.LES becomes computationally costly at city scale.Source: §1 · p. 3
• Most existing DNNs learn mappings between finite-dimensional spaces and can struggle to generalize when governing-equation parameters or boundary conditions change.Source: §1 · p. 4
• Li et al. proposed the Fourier Neural Operator (FNO) for learning the Navier–Stokes equations, parameterizing it in Fourier space to approximate PDEs accurately and train faster than existing DNNs [42].The introduction attributes its efficiency to representing continuous PDE inputs and outputs in Fourier space and to the equivalence between Euclidean-space convolution and Fourier-space pointwise multiplication.Source: §1 · pp. 4–5

• Pangu-Weather [56] outperformed traditional numerical weather prediction methods in accuracy and speed, while GraphCast [57] outperformed HRES and prior ML forecast models in most evaluated cases.Source: §1 · pp. 5–6
• ClimaX [58] addresses generalization by fine-tuning a pretrained model across climate and weather tasks and training on diverse datasets.Source: §1 · p. 6
• FengWu [59] uses a multimodal, multitask architecture and extends high-accuracy global medium-range forecasting to lead times of 10.75 days.Source: §1 · p. 6
• FNO demands the least graphics memory for training among the transformer and GNN models used in the cited weather-forecasting approaches.Source: §1 · p. 6
• The paper demonstrates a highly accurate 3D wind prediction model trained using just one Tesla A100 GPU.Source: §1 · p. 6

2.1 CityFFD simulation and data description

• CityFFD generates the urban CFD data using semi-Lagrangian and fractional-step methods on GPUs, with Lagrangian advection, LES turbulence modeling, and fourth-order interpolation.The solver was validated against CFD benchmarks for isothermal and non-isothermal conditions and against onsite wind-speed measurements.Source: §2.1 · pp. 6–7

2.2 Data pre-processing

• The dataset comprises 1,200 Niigata wind-field time steps at 0.1-second intervals, segmented into 598 six-step samples with five inputs and one output.A stride of two time steps is used for the sliding window.Source: §2.2 · p. 9
• The 598 samples are randomly split into 500 training and 98 testing examples, retaining transient spin-up data to assess performance beyond stable flow.The simulation begins with a west wind direction, and the full dataset spans the initial state through a relatively stable state.Source: §2.2 · p. 9
• To fit GPU memory, third-order spline interpolation downsamples the 400 × 400 × 150 velocity-magnitude grid to 200 × 200 × 150 while preserving data details.The paper notes that the original grid requires substantial GPU memory and that the downsampled input fits one training sample on a Tesla A100 GPU.Source: §2.2 · p. 9

2.3 FNO model construction

• The FNO lifts inputs with a fully connected layer, applies shape-preserving Fourier layers, then projects outputs back to the target dimension.The lifting and projection layers use 5 × 20 and 20 × 1 real-valued weight matrices, respectively.Source: §2.3 · p. 11, Figure 3 · p. 11
• The model receives a 200 × 200 × 150 × 5 input tensor and lifts it to 20 channels using a fully connected layer with a 5 × 20 real-valued weight matrix.The first three dimensions are the spatial grid and the last dimension represents sequential time steps.Source: §2.3 · p. 12
• Each Fourier layer uses an encoder-decoder sequence of Fourier transform, linear transformation of truncated modes, inverse Fourier transform, and nonlinear activation.The Fourier-layer structure is shown in Figure 4.Source: §2.3 · p. 12, Figure 4 · p. 13
• In the first Fourier layer, the 20-channel tensor is permuted before transformation, truncated to 8 × 8 × 8 modes, weighted by a complex-valued tensor, zero-padded, and inverse-transformed.The nonlinear activation is applied after the inverse Fourier transform.Source: §2.3 · p. 13

3.1 Model training

• Because zero absolute velocity magnitude occurs at certain grids, typical grid-wise relative error cannot be directly computed; therefore, average layer-wise relative error is adopted as the loss function.Source: §3.1 · p. 13

3.2 Testing performance : one-step prediction and generalization ability

• The west-wind test loss was 0.3% when the model was trained on west-wind data and given five continuous time steps to predict the next step.The five-step input was chosen as a trade-off between memory consumption and prediction accuracy.Source: §3.2 · pp. 14–15
• For a model trained only on west wind, test loss rose to approximately 3% for north and south winds and around 5% for east wind.The authors describe these errors as acceptable, while noting that prediction error tends to increase for different wind directions.Source: §3.2 · p. 15
• At 2 m, west-wind prediction errors were minute, but larger errors appeared near building edges and where wind velocity changed.The paper attributes edge errors to difficulty capturing boundary information from training data alone; errors were also larger in open spaces for north, south, and east winds.Source: §3.2 · pp. 15–16, Figure 6 · p. 16
• Predicted velocity-magnitude distributions closely matched ground truth for west, north, and south winds, while east wind showed the largest gap across nearly 0–9 m/s.The velocity inside buildings was defined as zero and included in the distributions.Source: Figure 8 · p. 17, §3.2 · p. 18
• Conditional absolute error was higher at both low and high velocities across wind directions, with west wind lowest and east wind highest.The authors link extreme-velocity errors to the data-driven FNO's lack of strict boundary constraints.Source: §3.2 · p. 18, Figure 9 · p. 18
• Across all wind directions, predicted velocities closely aligned with ground truth as height varied.A further visualization evaluated one-step predictions after training with both west and south wind data.Source: Figure 10 · p. 18, §3.2 · p. 19, Figure 11 · p. 19
• North-wind test loss fell from 3% to 2%, east-wind from 5% to 2%, and south-wind from 3% to 0.3% after adding south-wind training data.The resulting south-wind loss was similar to west-wind loss; west-wind loss did not further decrease with data from other directions, suggesting performance there had reached a limit.Source: §3.2 · pp. 19–20

3.3 Testing performance : sequential time step prediction

• Sequential predictions accurately reconstructed initial velocity fields, but accumulated errors grew in magnitude and spatial extent over time.The reported pattern applied to horizontal slices and 3D velocity-magnitude fields.Source: §3.3 · pp. 21, 23
• About 15 steps brought accumulated absolute error to 0.1 m/s, and it reached 1 m/s within 70 steps; growth was approximately linear for 60 steps, then nearly exponential.Figure 16 presents accumulated error over 70 sequential steps.Source: Figure 16 · p. 23, §3.3 · p. 23

3.4 Computational efficiency comparison

• The FNO approach provides over 25 times speedup compared with the traditional numerical solver, allowing real-time 3D dynamic simulations.Table 2 reports computational efficiency performance.Source: §3.4 · p. 23, Table 2 · p. 24

4 Conclusion

• The work extends FNO to high-resolution urban-scale fluid simulation, a setting where prior 3D FNO applications were limited to grids of 64 × 64 × 64.The authors describe their adaptation to extensive data as one of the pioneering FNO efforts.Source: §4 · p. 24
• Because high-resolution 3D wind-field data can exceed GPU memory, the study uses cubic-spline downsampling from 400 × 400 × 150 to 200 × 200 × 150 grids for training on a Tesla A100.The authors state that this reduction retains essential details.Source: §4 · p. 24
• The authors suggest the downsampling approach may benefit other scientific-computing tasks involving high-resolution data.Source: §4 · p. 24
• The FNO significantly expedites 3D urban wind-field simulation while reconstructing instantaneous spatial velocity magnitude with an error rate below 1%.The reported accuracy concerns the instantaneous spatial velocity magnitude field.Source: §4 · p. 24
• Trained solely on data from one wind direction, the FNO generalizes well to distinct wind directions.Source: §4 · p. 24

5 Discussion and future work

• The current FNO predicts only absolute wind speed because high-dimensional data and constrained GPU memory limit the model.Future work aims to develop more efficient, lightweight models and predict additional variables, including velocity components.Source: §5 · p. 25
• Although generalization across wind directions has been tested, training uses fixed speed and direction, and future work targets other speeds and cities.The authors describe real-world urban microclimate simulations as more complex and challenging.Source: §5 · p. 25
• Training data use constant wind speed and direction, so performance under dynamic wind fields remains to be investigated.The authors note that real-world wind fields have chaotic speed and direction.Source: §5 · p. 25
• Prediction error increases and accumulates over time in the chaotic dynamic system; proposed remedies include attention and physical constraints.The physical constraints would align predictions with governing equations and conservation laws.Source: §5 · p. 25
• Future training will include building size, density, and spacing as morphological inputs to improve the model’s physical understanding and generalization.The authors expect these parameters to support a more comprehensive understanding of physical relations intrinsic to the environment.Source: §5 · p. 25