# Multi-Fidelity Bayesian Optimization of Metasurface Designs

Ayoub Bellouch, Mahmoud Elsawy, Guillaume Leroy, Mickael Binois, Régis

Duvigneau, Stéphane Lanteri

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Ayoub Bellouch, Mahmoud Elsawy, Guillaume Leroy, Mickael Binois, Régis Duvigneau, et al. Multi-Fidelity Bayesian Optimization of Metasurface Designs. EMTS 2025 - 25th URSI International Symposium on Electromagnetic Theory, URSI Commission B, Jun 2025, Bologna, Italy. hal-05288409

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# Multi-Fidelity Bayesian Optimization of Metasurface Designs

A.Bellouch*(1), M. Elsawy(1), G. Leroy(1), M. Binois(1), R. Duvigneau(1) and S. Lanteri(1)

(1) Université Côte d'Azur, Inria, CNRS, LJAD, 06902, Sophia Antipolis Cedex, France

## Abstract

This paper proposes a multi-fidelity (MF) modeling approach for optimizing advanced metasurface deflector designs. By integrating low- and high-fidelity full-wave electromagnetic simulations, our efficient global optimization (EGO) algorithm, based on a non-nested recursive MF model, achieves faster convergence while significantly reducing dependence on costly high-fidelity evaluations. A comparative analysis with the conventional single-fidelity EGO optimization approach demonstrates that our method achieves comparable performance in terms of design diffraction efficiency, with the added benefit of significantly reducing the computational cost. This study highlights the potential of this methodology to accelerate design processes while maintaining accuracy. This makes it a practical solution to challenging electromagnetic propagation optimization problems that require significant computational resources.

## 1 Introduction

Electromagnetic (EM) wave propagation problems can be modeled with varying levels of fidelity, where the accuracy of the numerical analysis is influenced by several critical simulation parameters. These factors, such as mesh size, boundary conditions, and specific solver parameters, not only determine the accuracy of the model but also directly influence computational cost. Recently, to optimize the performance of complex EM propagation systems based on metasurfaces, we have opted for one of the most advanced techniques belonging to the class of Bayesian optimization (BO) methods, known as efficient global optimization (EGO) [1]. It employs a cheaper surrogate model to replace the objective function, providing significant benefits in the management of expensive simulations [2]. This surrogate model uses a statistical learning criterion, called the acquisition function, related to the optimization objective to identify which designs should be tested in each iteration of the optimization process. Although the EGO method requires fewer solver calls to converge compared to traditional global optimization algorithms, each full-wave simulation is significantly time consuming due to the numerical complexity involved in modeling metasurfaces [3]. The multi-fidelity (MF) approach has been introduced to address this challenge, enabling the construction of an autoregressive model (AR) [4] that incorporates observations

from models with different levels of fidelity. In [5], the authors apply the AR model to MF optimization within the EGO algorithm framework. Subsequently, Le Gratiet and Garnier [6] have proposed a recursive formulation of the MF model, which allows the determination of the posterior surrogate model for intermediate levels and significantly reduces computational costs. Since the introduction of MF models in EGO optimization, constructing an acquisition function for this framework has proven challenging, primarily due to the complexities involved in simultaneously determining the next design and the next level of fidelity to evaluate. As a result, only a few strategies have been pursued, including extensions of standard acquisition functions, such as the expected improvement (EI) [7] and the knowledge gradient (KG) [8], alongside the implementation of novel, efficient information-theoretic BO acquisition functions for MF and multi-task (MT) frameworks, such as Multi-task Max-value Bayesian Optimization (MUMBO) [9]. The advantages of the MUMBO approach are its ability to handle both continuous and discrete fidelity spaces, and its low computational cost for hyper-parameter tuning tasks. In this work, we leverage the recent advancements in MF frameworks along with the MUMBO acquisition functions to make the EGO algorithm more efficient for optimizing sub-wavelength metasurface designs. Using a two-level fidelity approach, we improve the learning of the GP model. This enhancement not only helps the objective function converge more efficiently but also significantly reduces the computational cost.

## 2 Design methodology

### 2.1 Optimization strategy

In this study, we employ the recursive formulation to efficiently construct the s-level GP surrogate models. Let $\mathcal{Z}^s = (Z^s(x))_{x \in \mathbb{R}^d, \ell=1,...,s}$ be the Gaussian process modeling the s levels of fidelity, the recursive inference strategy for MF model :

$$ \left\{ \begin{array}{l} Z^\ell(x) = \rho_{\ell-1} \hat{Z}^{\ell-1}(x) + \delta^\ell(x) \\ \hat{Z}^{\ell-1}(x) \perp \delta^\ell(x) \\ \hat{Z}^{\ell-1}(x) \sim [Z^{\ell-1}(x) | Y_{n_{\ell-1}}^{\ell-1}] \end{array} \right. \quad \ell = 2, ..., s \quad (1) $$

where $\hat{Z}^{\ell-1}(x)$ is the posterior distribution of the GP prior $Z^{\ell-1}(x)$ constructed from the observations $Y_{n_{\ell-1}}^{\ell-1}$, $\rho_{\ell-1}$ is a
regression parameter and $\delta^\ell(x)$ is a Gaussian process in-dependent of $(Z^\ell(x))_{\ell=1,...,s}$. The multi-fidelity Gaussian posterior distribution of the level of fidelity $\ell$ predicted by the recursive scheme for non-nested observations is given by :

$$[Z^\ell(x) | Y_{n\ell}^\ell] \sim \mathcal{N}(m_\ell(x), \sigma_\ell^2(x)) \quad (2)$$

with mean function :

$$m_{\ell}(x) = k_{\ell}^{T}(x) (C_{\ell}(\theta_{\ell}) + \sigma_{\delta_{\ell}}^{2})^{-1} (Y_{n\ell}^{\ell} - \rho_{\ell-1} m_{\ell-1}(x)) \\ + \rho_{\ell-1} m_{\ell-1}(x) \quad (3)$$

and variance :

$$\sigma_{\ell}^{2}(x) = \sigma_{\delta_{\ell}}^{2} - k_{\ell}^{T}(x) (C_{\ell}(\theta_{\ell}) + \sigma_{\delta_{\ell}}^{2} I_{\ell})^{-1} k_{\ell}(x) \\ + \rho_{\ell-1}^{2} \sigma_{\ell-1}^{2}(x) + \kappa_{\ell}(x) \quad (4)$$

where $C_\ell(\theta_\ell) \in \mathbb{R}^{n_\ell \times n_\ell}$ is the covariance matrix of the Gaussian vector $\mathcal{X}$, $k_\ell(x)$ is the covariance vector between $Z^\ell(x)$ and $\mathcal{X}$, $\kappa_\ell(x)$ is the variance of $Z^\ell(x)$, $\sigma_{\delta_\ell}^2$ is the noise variance and $\theta_\ell$ are the set of hyper-parameters of a the $\ell$ fidelity level. This approach significantly reduces computational complexity compared to the AR model, which becomes computationally intensive as the number of observations increases because of the need for covariance matrix inversions involving all fidelity levels. Additionally, it offers access to the predictive mean and variance at intermediate levels, ensuring consistent predictions across fidelity levels while enabling separate computations at each level.

To guide the optimization process, a statistical learning criterion based on the MUMBO acquisition function is used. This function selects the next point and the level of fidelity that offers the highest information gain per unit of cost :

$$(x_{n+1}, \ell_{n+1}) = \operatorname*{argmax}_{(x, \ell) \in \mathcal{X} \times \mathcal{N}} \frac{\alpha_n(x, \ell)}{c(x, \ell)} \quad (5)$$

where $\alpha_n: \mathcal{X} \times \mathcal{N} \to \mathbb{R}$ is an extended acquisition function and $c: \mathcal{X} \times \mathcal{N} \to \mathbb{R}^+$ is a cost function. Extend the efficient information-theoretic acquisition function that seeks to reduce our uncertainty in the max value $g^*$ of the objective function. MUMBO seeks to compute the information gain [9]:

$$\alpha_n^{\text{MUMBO}}(x, \ell) = H(y(x, \ell)|D_n) - \mathbb{E}_{g^*} [H(y(x, \ell)|g^*, D_n)] \quad (6)$$

where $H$ is the differential entropy, which measures the uncertainty.

## 2.2 Metasurface design

We apply the MF optimization framework to optimize the metasurface deflector design proposed by Elsawy et al. in [10]. We consider a phase gradient metasurface deflector (see Figure 1) made of rectangular GaN semiconductor (dark red regions) placed on a semi-infinite substrate

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Design of the periodic macrocell of the phase gradient meta-deflector.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b)</figcaption></figure>

<caption>**Figure 2.** Top view of the deflector simulation: (a) design with low fidelity mesh (b) design with high fidelity mesh.</caption>

made of Al₂O₃ (shown in green). The objective is to optimize the all-dielectric cylindrical nanopillars to maximize the diffraction efficiency in the first order mode. Due to the symmetry of the nanopillars, we are targeting polarization insensitive properties. For this example, we optimized the structures by modifying 7 parameters including the pillars diameter, the height and the position of the nanopillars. To

<table><caption>**Table 1.** Configuration of the fidelity levels</caption><thead><tr><th>Fidelity</th><th>Ordre</th><th>Energy</th><th>Mesh size</th><th>Mesh ratio</th></tr></thead><tbody><tr><td>LF</td><td>P<sub>2</sub></td><td>0.01%</td><td>* Coarse</td><td>3</td></tr><tr><td>HF</td><td>P<sub>2</sub></td><td>0.01%</td><td>* Standard</td><td>1</td></tr></tbody></table>

model the metasurface deflectors and, more importantly, to effectively capture the strong near-field coupling generated by the elements, a full-wave approach was utilized to simulate the deflector by solving the complete system of time-domain Maxwell equations. For this purpose, we exploit the Discontinuous Galerkin Time-Domain (DGTD) solver that was proposed in [11] and which is implemented in our in-house simulation software called DIOGENES [12]. In this study, we consider an optimization model with two levels of fidelity. The selection of these two levels, as illustrated in Figure 2, is based on the mesh criterion (Figure 1). Table 1 presents the simulation configurations for the two levels of fidelity.
## 3 Numerical results

The aim of the optimization is to find the most efficient arrangement and size of all-dielectric nanopillars to maximum light deflection efficiency $\tilde{\eta}(0, +1)$ of the mode (0, +1) at a wavelength of $\lambda = 600$ nm. The parameters $X$ to be optimized include the diameter $D_{i} \in [1,4]$, along with $\rho_{i} \in [1,3]$ the distance between elements within the macro-cell, leading to 7 parameters of optimization. To investigate how the objective function changes between the two fidelity levels, we conducted a parametric study focusing on efficiency $\tilde{\eta}(0, +1)$ as a function of the first pillar's diameter $D_1$, keeping the remaining parameters constant. Figure 3 illustrates a strong linear relationship between the fidelity levels, with similar optimal variations observed. This demonstrates the potential to gain meaningful knowledge about the high fidelity model from low-fidelity simulations.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** The efficiency in function of $D_1$[nm].</figcaption></figure>

The performance of the MF-EGO algorithm is evaluated against the classical single-fidelity (SF) EGO optimization. To handle a minimization problem, the objective function is expressed as $J(\mathbf{X}) = 1 - \tilde{\eta}(0, +1) \in [0, 1]$. During the first phase, which is referred as DOE, an equivalent database is generated with 75 points: (a) 75 high-fidelity simulations for the SL-EGO method; (b) 50 low-fidelity and 25 high-fidelity simulations for the MF-EGO method. In the second phase, 150 iterations are used for both SL-EGO and MF-EGO to refine the GP surrogate models during the optimization process and determine the global minimum. The cost ratio of the MUMBO acquisition function for the MF-EGO is set to 8 or 10, based on the computational time ratio between the HF and SF simulations. The EGO optimization for both approaches was performed on a server with 50 cores.

Figure 4 summarizes the optimization process for both EGO approaches with cost ratio of 8 (Figure 4.a) and cost ratio 10 (Figure 4.b). The objective function evaluations

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Evolution of the objective function with the number of full-wave solver calls for simple-fidelity EGO and multi-fidelity EGO: (a) Cost ratio: 8, (b) Cost ratio: 10</figcaption></figure>

from the SF-EGO are represented by black points, while the evaluations from the MF-EGO are shown with blue points for LF simulations and red points for HF simulations. The SF-EGO algorithm required 17 hours and 36 minutes to converge, including a DOE phase that lasted 6 hours and 20 minutes. The best design achieved an efficiency of 85.4%. In comparison, our MF-EGO approach demonstrated much faster convergence. The DOE phase was completed in 1 hour and 27 minutes for a ratio of 8 and in 1 hour and 34 minutes for a ratio of 10. The EGO phase further reduced runtime, converging in 6 hours for a ratio of 8 and in only 4 hours and 18 minutes for a ratio of 10. The best designs obtained by MF-EGO achieved, respectively, efficiencies of 84.8% and 84.6%. These results demonstrate the effectiveness of our approach in significantly reducing computational time while maintaining comparable performance.

By analyzing the impact of the cost ratio, we observe that varying this parameter introduces flexibility depending on the optimization strategy. With a cost ratio of 8, the EGO phase performed 43 LF simulations and 107 HF simulations. This configuration allowed LF simulations to accelerate
erate convergence early in the process, while the MUMBO
acquisition function later focused on HF simulations to ex-
ploit the region around the minimum. Conversely, for a ra-
tio of 10, the algorithm performed 113 LF simulations and
only 37 HF simulations, prioritizing exploration of new re-
gions and significantly reducing computation time.

4 Conclusion

This work highlights the advantages of multi-fidelity mod-
eling in optimizing advanced metasurface deflector designs.
The ability to achieve rapid convergence while minimizing
reliance on costly high-fidelity simulations makes our re-
cursive MF-EGO algorithm computationally efficient com-
pared to the traditional simple-fidelity EGO approaches.
The flexibility of the cost ratio further enhances its adapt-
ability, enabling an effective balance between exploration
and exploitation. It also facilitates the capture of high-
performance metasurface designs while ensuring efficient
computational cost management. These results illustrate
the potential of Multi-fidelity BO as a robust and computa-
tionally efficient solution for addressing sophisticated EM
propagation optimization problems.

5 Acknowledgements

This work was co-funded by the Defense Innovation
Agency (AID) of the French Ministry of the Armed Forces
as part of the RAPID AEROCOM project No. 2022 29
0954.

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