## 2 Theory
### Genetic Algorithm
Furthermore, the GA's ability to perform a global search without relying on gradient information allows it to explore the solution space more broadly and avoid getting trapped in local minima. This makes it particularly useful for more complex configurations with potential multiple minima or artefacts, where traditional methods might need help or overlook optimal solutions. Therefore, developing a different complementary approach such as GA can become increasingly valuable, especially in more sophisticated scenarios, and enhance our ability to solve and understand liquid crystal configurations. One of the main challenges in applying genetic algorithms or other optimization techniques to high-dimensional optimization problems is the exponential increase of com  
Figure 2: (**a**) General Genetic Algorithm Scheme. (**b**) Specific conducted GA Routine’s Scheme. putational resources with for a higher number of variables, a phenomenon known as the \"curse of dimensionality\". The results section demonstrates that this problem is mitigated through our PCA-based lower-dimensional remapping method. This genetic algorithm finds application in determining LC director configurations that minimise Frank-Oseen free energy. More specifically, the GA method here proposed considers directly Equation (2) instead of the one related to the application of the Euler-Lagrange Equations (see Equations (5) and (6) in [8]). Figure 2b presents its more precise routine schematic. As a start, a uniform random population of the lower-dimensional remapped solution space is created, representing the remapped theta and twist three-dimensional space, which trivially reconstructs the unit nematic director vector distribution using the proper trigonometric transformations. The GA optimisation is then executed with 3500 maximum generations, population sizes of 60 individuals and mutation standard deviations (Gaussian Mutation) of 0.003, and the voltage is updated solving the Gauss's Law following the procedure defined in [8; 31]. The population is reinitialised as a Gaussian distribution around the former best solution. This process is repeated seven times until the convergence criterion is met, with decreasing standard deviations in each new GA optimisation cycle for the population's re-initialisation. Finally, theta-space-only and twist-space-only optimisation refinements are conducted, following the same logic but with smaller population sizes of 10 individuals, smaller mutation standard deviations (0.0003) and 10,000 maximum generations. The fraction of individuals used for crossover (Crossover Arithmetic function) is set to be 20% during optimisation. The function used to select the fittest individuals is the standard Stochastic Universal Sampling, a variation of the fitness-proportionate selection method, with three best 'elite' individuals preserved for the next generation. The decision to update the voltage only once for each GA optimisation iteration is strategically motivated by computational efficiency and empirical observations. The computational time cost associated with voltage updates for each individual in each algorithm generation is an obvious and relevant consideration, especially in high-resolution optimisation processes. Moreover, practical experience has revealed that infrequent voltage updates significantly reduce the likelihood of the genetic algorithm (GA) getting stuck in local minima, as it prevents the algorithm from oscillating around suboptimal solutions. This approach, along with the PCA-based lower-dimensional remapping, enhances the time costs and robustness of the optimisation process.