J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026) 439
Figure 1. Example of inscription matrix on a sol-gel sample.
Grey-Box( 8): the value of an intermediate quantity( the color) is leveraged here, contrary to typical Bayesian optimization functions that aim at predicting the objective function directly.
Thanks to its probabilistic nature, the BO weights both exploration and exploitation by targeting areas that have already been identified as performing( exploitation) or areas with high uncertainty( exploration). This approach prevents the algorithm from becoming trapped in a certain region, which is a common issue with GA. It is important to note the structural difference in the problem formulation between the two algorithms. The objective being to maximize V ðPÞ the volume of( the convex hull of the colors produced by) a set of parameters, standard GA and BO need to be adapted. Indeed, if one considers the complete set P as the input space for GA or BO, then it means many laser inscriptions are to be done at each iteration.
For GA, it relies on individuals P i to perform mutations and crossovers, and relies on individual fitness for selection. Because the volume is a property of a set, not an individual, the challenge is to design an individual fitness function that eventually correlates with maximizing the collective volume. For this purpose we followed Cucerca et al. [ 30 ] and converted the problem to a multi-objective one. The fitness of an individual is evaluated with respect to the current population based on its contribution to hue diversity( in the CIE LCh color coordinates) and saturation. This multi-objective framework is necessary to maintain a diverse Pareto front; a standard single-objective GA would quickly converge to a single dense region of the color space, failing to span a wide gamut.
For BO, as the approach is more flexible, we propose a new grey-box approach( i. e. where part of the objective function is known). The surrogate model( GP) models the intermediate physical property by predicting the color of individual points, but the objective function is evaluated at the set level. The acquisition function analytically estimates the expected gamut improvement by averaging over all possible outcomes, defined as the added volume a predicted color brings to the existing set( Eq.( 5)). This objective function has the added benefit of direct applicability in scenarios involving multimodes colors. It has been shown already that predicting the resulting spectra obtained using any processing parameter set for those metasurface was doable [ 26 ] using a deep neural network. However our approach aims to reach the optimal gamut configuration as rapidly as possible while relying on a limited number of experimental data points, making Bayesian
optimization more adapted than deep learning model that requires a large amount of data and will not adapt easily to a new initial material.
A gamut hypervolume can then be computed even for higher dimension( 6D for 2 modes, 9D for 3 modes, etc.), which makes it easy to introduce the concept of image multiplexing.
The first two steps are the same as with GA. The last step looks similar as well, but it is here possible to only reinscribe one color in a new iteration, unlike the GA that required to inscribe a batch of colors each time.
The objective of the proposed framework is to maximize the diversity of structural colors generated on the material surface by intelligently exploring the available laser settings. The input space X R 3 is strictly bound by the physical limits of the laser system: power( 40 – 457 mJ / cm 2), scanning speed( 50 – 2000 mm / s), and repetition rate( 10 – 600 kHz).
" i, p i 2 X is a parameter vector with a corresponding output that is the 3-dimensional color coordinate vector c i ¼ colðp i Þ 2 R 3 in the CIELAB color space.
Hypervolume V( P) is computed as the volume of the convex hull of the colors obtained from the finite vector parameter set P ={ p 1, p 2,...} intheL * a * b * space. We choose this space for defining both the state space and the objective function due to its perceptual uniformity.
The hypervolume associated to a set of laser parameters can be expressed as:
VðPÞ: ¼ V ConvexHull fcolðp i Þ: ð1Þ g pi 2P
Formally, for a fixed size n, the problem is to find the optimal set of laser parameters P n that maximizes the gamut hypervolume V, which is defined as the volume of the convex hull formed by the colors induced by P *, namely:
P n ¼ argmax P2X n VðPÞ:
However, because the true physical mapping col( p) is a highly complex unknown function, finding the optimal set P * is unrealizable. Instead, the optimization must be performed sequentially in order to acquire information about col, the function linking the laser parameters and the produced color on the current sample. In its sequential form, where the set of parameters is constructed one point after the other and the true color is obtained, the problem can be written, at each iteration n as:
P nþ1 ¼
p nþ1 [ P
n
However, as pointed out, the true physical color col( p) isa black-box outcome before the actual physical experiment. Consequently, the physical volume equation cannot be solved directly. The optimal sequential solution P n actually ignore the colors measured( as it supposed the col function is known), but any algorithm will use the measured colors C n as a( very partial in the first iterations) information about the col function and thus will fail to achieve the optimal. The surrogate model consists in a Gaussian Process( GP), that is fitted using the( n � 1) data points and is fully ð2Þ
ð3Þ