The work is a computational demonstration based on simulation data, not a field trial, and the preprint is currently under review in Wind Energy Science. The reference simulations used in the study covered terrain domains as large as 3 by 3 kilometres at 30-metre resolution and represented the flow around solar installations and wind turbines at 0.5-metre resolution. Obtaining these fields with a conventional computational fluid dynamics calculation is expensive, particularly when many candidate configurations must be considered.

The proposed surrogate is intended to address that computational cost by producing full fields rapidly after it has been trained against OpenFOAM results. The first architecture tested was a point-wise hybrid model combining a coordinate multilayer perceptron with two-dimensional convolutional terrain encoders. The second was a three-dimensional grid-based U-Net.

Both predict the local wind velocity and kinematic pressure rather than a single summary value for a site. The two families were designed as two-stage cascades: an initial model estimates the broad terrain-scale background, and a second stage refines the prediction around the structures. Both cascades were trained as physics-regularised surrogates, meaning that their losses combined disagreement with the CFD reference data with penalties for departures from specified physical and numerical conditions.

Those penalties covered mass conservation, momentum conservation, the prescribed inflow profile, outlet pressure, and impermeable lateral and top boundaries. This training arrangement links the learned predictions to the governing conservation requirements and boundary conditions represented in the reference setup. It does not mean that the networks solve the full CFD equations from first principles; the study evaluates them as learned surrogates trained from simulations, with additional penalties for violations of selected constraints.

With the full training budget of 292 domains, the final models produced normalised wind-speed root mean square errors of approximately 0.16 over the global background domain and 0.17 within regions of interest. The two architectures reached comparable aggregate values. Pressure was more difficult over the full domain, where the normalised root mean square error was about 0.24, while the corresponding regional value was around 0.12.

The authors also address a definitional ambiguity in kinematic pressure, which is physically determined only up to a constant. Removing the arbitrary offset for each case with a gauge correction lowered the global pressure error to roughly 0.18–0.20 and the regional error to about 0.09. Matched comparisons between physics-regularised and data-only models indicated that the clearest contribution of the additional penalties was consistency with mass and momentum conservation rather than a decisive improvement in supervised accuracy.

Differences in wind-speed error were within the spread between random training seeds, so the study did not establish a distinct accuracy advantage on that measure. The physical residuals showed more direct effects: continuity and momentum residuals were approximately halved in regions containing structures and were reduced by 15% to 65% at terrain scale when physics penalties were included. In other words, the regularised models better respected the conservation behaviour encoded in the training setup while retaining broadly similar agreement with the CFD data.

The researchers also used active learning to expand the simulation dataset. Beginning with 120 domains, they added simulations according to a score-based selection procedure until the training set contained 292 domains. This expansion reduced terrain-scale test error by about 30%, with the normalised wind-speed root mean square error decreasing from 0.26 to 0.18.

At a matched training budget, the score-based selection strategy outperformed random selection. The reported result supports targeted acquisition of additional simulations within this particular setup, but the training process still depends on the availability of CFD outputs and does not demonstrate that new simulations will no longer be necessary. Although their aggregate error levels were similar, the two architectures showed a practical trade-off.

The point-wise hybrid model attained the best aggregate errors and could evaluate flexible point locations, allowing selected outputs to be obtained without requiring only a complete grid prediction. The three-dimensional U-Net better resolved channelling jets between closely spaced structures. Such narrow flow features were scarcely registered by volume-averaged metrics, which could therefore make the models appear more alike than their spatial outputs actually were.

The comparison does not establish one architecture as universally superior: the hybrid performed better in aggregate error, while the U-Net captured specific fine-scale structures more effectively. The main practical attraction is the time required to obtain a prediction. Once trained, the surrogate returns complete wind and pressure fields in seconds, making rapid micro-siting comparisons, alternative layout evaluation and preliminary loading studies feasible without launching a new high-resolution CFD calculation for every iteration.

The results do not show that the system can replace OpenFOAM for every application. The study is limited to steady-state RANS reference simulations over complex Swiss terrain and the specific solar and wind-energy structures represented in the dataset, and its performance was assessed against those simulations rather than an equivalent body of field measurements. The preprint remains open for discussion and has not yet completed journal peer review.

Within those limits, the study presents a practical division of labour between high-cost simulation and rapid inference. OpenFOAM simulations provide the reference data from which the surrogates learn, while score-based selection identifies cases for further acquisition. After that development stage, the trained networks can explore many fine-scale scenarios in seconds.

The central measured finding is not that physics penalties made the surrogate uniformly more accurate, but that they improved conservation consistency with little clear change in wind-speed accuracy, while active learning improved terrain-scale test performance and two complementary architectures exposed different spatial strengths. Together, these results suggest that physics-aware neural surrogates could accelerate renewable-energy layout exploration while preserving the detailed three-dimensional fields needed for engineering evaluation.

Subject of Research: Neural-network surrogates for wind and pressure fields over complex terrain and renewable-energy structures

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

Article References: Gasser, J., Frischholz, Y., & Lehning, M. (2026). Physics-regularised neural network surrogates for fine-scale wind and pressure fields over complex terrain and structures. https://doi.org/10.5194/wes-2026-179

Image Credits: AI Generated

DOI: 10.5194/wes-2026-179

Keywords: neural-network surrogates, computational fluid dynamics, wind flow, pressure fields, complex terrain, renewable-energy siting, wind turbines, photovoltaic structures, physics-informed learning, active learning, Physics-regularised, neural