Optimal Transport-based Permutation-Invariant Bayesian Optimization of Offshore Wind Farm Layouts

Antonio Candelieri; Laurens Bliek

cs.AIMar 27, 2026

Optimal Transport-based Permutation-Invariant Bayesian Optimization of Offshore Wind Farm Layouts

Authors: Antonio Candelieri, Laurens Bliek

Organizations: Department of Economics Management and Statistics, University of Milano-Bicocca, Milan, Italy. · Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, Eindhoven, the Netherlands. · Eindhoven AI Systems Institute, Eindhoven, the Netherlands.

Abstract

Bayesian Optimization (BO) is widely and successfully adopted for solving optimization problems having an expensive-to-evaluate, black-box, and non-convex objective function. However, the vanilla BO algorithm is not able to exploit possible symmetries characterizing the target problem. An intuitive case is given by optimal location problems, whose decision variables refer to a finite set of points within a continuous space, with the order of points not affecting the value of the objective function. We refer to this setting as optimization over layouts to distinguish from optimization over point-clouds where, instead, the order of points counts. As an instance of optimization over layouts we consider a real-life industrial-relevant application, that is the optimization of the layout of an offshore wind farm: given identical wind turbines, switching any pair of them has not any effect on the annual energy production. Based on Optimal Transport theory, we propose a Permutation-Invariant BO approach, namely PIBO, proved to provide better wind farm layouts when compared to the vanilla BO approach while cutting computation time roughly in half.

Figures & tables

Panel of Figure 1 — Figure 1: A simple example of the undesired replication: on the left, the Bird test function (with search space rescaled in $[0,1]^{2}$ and changed in sign for considering maximization); on the right, a function over layouts (aka permutation-invariant function) obtained by mirroring the original Bird function from below to above the bisector (i.e., $f(x_{1},x_{2})=f(x_{2},x_{1})$ ). Although the true function over layouts is just the one below the bisector, with just two global optima, optimizing over point-clouds, like in formulation 1 and 2 , leads to consider a function with four global optima (i.e., replication mechanism).

Baseline	$y_{best}$ avg (sd)	% feasible solutions
LHS in the Physical Space	69.27 (0.63)	94%
LHS in the Flow Space	68.39 (1.24)	56%
URS in the Physical Space	68.71 (1.24)	51%
TPE in the Physical Space	69.75 (0.62)	57%

Table 1: Value of the optimal solution (average and standard deviation over 30 independent runs) provided by the baseline approaches (i.e., not relying on any probabilistic surrogate model): the higher the better. Percentage of feasible solutions (i.e., not violating the constraint) is also reported.

GP’s	$\xi$ in	PIBO (ours)	BO on flows	BO on point-clouds	runs won	runs won by
kernel	UCB	$y^{+}$ avg (sd)	$y^{+}$ avg (sd)	$y^{+}$ avg (sd)	by PIBO	BO on flows
Exp.	0	70.38 (0.32)	70.56 (0.41)	69.86 (0.50)	12/30	17/30
	1	70.60 (0.36)	70.46 (0.40)	69.95 (0.40)	16/30	13/30
	3	70.63 (0.38)	70.58 (0.38)	69.80 (0.37)	17/30	13/30
	6	70.77 (0.33)	70.66 (0.45)	69.82 (0.26)	15/30	15/30
Sq. Exp.	0	70.00 (0.39)	70.13 (0.52)	69.29 (0.66)	9/30	19/30

Table 2: Value of the optimal solution (average and standard deviation over 30 independent runs) for the three BO approaches: the higher the better. Best value for each pair kernel type -

\xi

value is in bold; the best overall is also underlined. Number of runs won by PIBO and vanilla BO on flows are also reported (runs won by vanilla BO on point-clouds is obtained by difference).

Figure 2: Comparing convergence of the three algorithms (PIBO, vanilla BO on flows, and vanilla BO on point-clouds) with Exponential kernel for the GP and $\xi=6$ in GP-UCB. Results are averaged on 30 independent runs. Important remark: for each run the three algorithms share the same set of initial solutions, randomly chosen via LHS within the physical space: vanilla BO on point-clouds keeps going by working in the same space while PIBO and vanilla BO on flows works on the space of flows. TPE do not share the same set of initial solutions because the hyperopt library – which implements it – only allows for uniformly random initialization in the physical space.

Panel of Figure 3 — Figure 3: Best solutions identified by the three BO algorithms, for the same run. GP’s kernel is Exponential and $\xi=6$ in GP-UCB. The gray square represents the physical search space (i.e., the area where wind turbines must be placed); the five points at the left-bottom corner represents the reference point cloud $\bar{P}$ (omitted in the rightmost picture because vanilla BO on point-clouds does not involve OT theory). PIBO (on the left) provides the highest value for the objective function and the associated solution is consistent with OT (i.e., no crossing flows). Vanilla BO on flows (in the middle) provides the second best solution among the three, but the placement is not consistent with OT (indeed, we observe crossing flows). Finally, vanilla BO on point-clouds (on the right) provides the worst solution among the three.

Panel of Figure 4 — Figure 4: Optimal wind farm layouts provided by the three BO implementations, separately. Each color is associated to one of the $m=5$ wind turbines, every point refers to the optimal location of that turbine for one of the 30 independent runs. Thanks to permutation-invariance at its core, PIBO is the only approach able to suggest a non-linear organization of the Physical Space into (possibly overlapping) influence regions , one for each wind turbine.

Bayesian Optimization

Bayesian optimization uses probabilistic models and sequential experiments to find strong solutions when evaluating a black-box objective is costly.

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State Key Laboratory of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China · School of Ocean Engineering and Technology, Sun Yat-Sen University & Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, 519000, China · Shenzhen Research Institute of Shanghai Jiao Tong University, Shenzhen, 518063, China+1

Surrogate Modeling Gaussian Process Regression Bayesian Optimization

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Wind farm layout optimization becomes increasingly difficult as turbine count grows, especially when high-fidelity aerodynamic models are used. Conventional Kriging-based Bayesian optimization often suffers from low sample efficiency in high-dimensional layout spaces because sparse training data provide limited domain coverage. We develop a Graph-informed Kriging framework that embeds reusable aerodynamic interaction information into the surrogate mean function. The wind farm is represented as a directed aerodynamic graph, where pairwise turbine interactions defined in a low-dimensional relative-coordinate space are assembled into a farm-scale mean. This structured mean captures the dominant variation in annual energy production (AEP), allowing the Gaussian process to model only the residual structure and uncertainty. Numerical experiments show that Graph-informed Kriging consistently improves AEP prediction accuracy over ordinary and generic-trend Kriging under sparse-data conditions, with increasing benefits as layout dimension and wake-interaction strength grow. Embedded in Bayesian optimization, it substantially improves sample efficiency and achieves competitive solutions for the IEA Wind Task 37 benchmark with about 280 AEP evaluations. In a controlled 16-turbine case, it also outperforms ordinary and multi-fidelity Kriging under the same high-fidelity evaluation budget. The framework is further extended to a 48-turbine wind farm, demonstrating applicability to large-scale CFD-based layout optimization. These results show that Graph-informed surrogate modeling can enable more efficient use of high-fidelity aerodynamic models for reliable large-scale wind farm layout design.

Abstract

Bayesian Optimization

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