Physics-regularised neural network surrogates for fine-scale wind and pressure fields over complex terrain and structures

Abstract. Detailed wind and pressure fields around structures in complex terrain are needed for renewable energy siting or structural load estimates, but Reynolds-averaged Navier-Stokes (RANS) simulations remain too expensive for repeated siting and design studies. This work develops a neural network based surrogate of steady-state OpenFOAM simulations that predicts the three-dimensional wind velocity and kinematic pressure distribution over terrain domains of up to 3 x 3 km size at 30 m resolution and, in a single framework, the flow and pressure fields around solar installations and wind turbines placed in this terrain at 0.5 m resolution. Two model families, a point-wise hybrid model combining a coordinate multilayer perceptron with two-dimensional convolutional terrain encoders, and a three-dimensional grid based U-Net, are compared. Both are designed as a two-stage cascade in which a terrain-scale background model is corrected by a structure-scale residual refiner. Both are trained as physics-regularised surrogates, i.e. not only on the CFD reference data but also with penalties on the violations of the mass and momentum conservation as well as on departures from the prescribed inflow profile, outlet pressure and impermeable lateral and top boundaries. With a training budget of 292 domains, the final models reach wind speed normalised root mean square error (RMSE) values of around 0.16 for global background and of around 0.17 for regions of interest, while pressure normalised RMSE is around 0.24 for global domain and 0.12 for regions of interest. Since the kinematic pressure is physically defined only up to a constant, removing the arbitrary per-case offset with a gauge correction reduces these errors to about 0.18–0.20 and 0.09 respectively. Both models reach comparable values. Matched data-only comparisons show that the contribution of the physics residuals is mass- and momentum-conservation consistency rather than supervised accuracy. The wind speed error differences lie within seed-to-seed spread, while the continuity and momentum residuals are roughly halved in the structures regions and reduced by 15 to 65% at terrain scale with physics residuals. Starting from 120 simulations, the training set is expanded to 292 domains by active learning, reducing the terrain-scale test error by about 30% (wind speed normalised RMSE from 0.26 to 0.18). At a matched budget, active learning score-based selection outperforms random selection. The two model families have a trade-off, where the hybrid attains the best aggregate errors and flexible point-wise inference, whereas the U-Net better resolves the channelling jets between closely spaced structures that volume-averaged metrics barely register. The surrogate returns full fields in seconds, making repeated micro-siting, layout comparison and preliminary loading studies practical.