## 3 Problem Description
### Modified Kernel Function  
The observation of permutation invariance is crucial to better acknowledging the principal idea introduced in this work. Since the black-box function of the BO framework (i.e., the simulator) considers the inputs to be sets, it would be extremely beneficial for this to also be incorporated in the BO framework. By sampling an input vector, information is obtained not only regarding what its objective function value is but also regarding all input vectors that are derived from same-group permutations of its elements. For instance, in the \(N\)-dimensional input space, where \(N=n+m\), and \(n\) and \(m\) correspond to the dimensions of the groups, a single sample would give information about \(n!\cdot m!\) vectors. It is now obvious that incorporating this information in BO would greatly benefit its efficiency. To incorporate this in BO, we need to recall that BO builds a surrogate objective function that is mostly influenced by samples and the covariance matrix which is derived from a kernel function. Modifying the kernel function is a clear pathway for improving the efficiency of BO. The kernel function calculates the covariance between points of the input space and acts as a measure of how similar the surrogate objective function's values are based on the distance between inputs. The Matern kernel utilized in this work is a stationary kernel, i.e., invariant to translations where the covariance of two input vectors is only dependent on their relative position \(k(\mathbf{x},\mathbf{y})=k(||\mathbf{x}-\mathbf{y}||)\). Inputs have high covariance when their distance approaches zero, whereas it decreases as they move further apart. The modified input space generated by the mapping of Algorithm 1 is useful for generating inputs inside the feasible region, but the caveat is that it is not homogeneous. Points that are close in the modified input space might be very far apart when translated to the original input space. The lack of preservation of this closeness property in the modified input space can be expressed as follows:  
\[\text{If }||\mathbf{c}_{1}-\mathbf{c}_{2}||\leq||\mathbf{c}_{1}- \mathbf{c}_{3}||\] \[\text{it does not necessarily imply that }||f(\mathbf{c}_{1})-f(\mathbf{c}_{2})||\leq||f( \mathbf{c}_{1})-f(\mathbf{c}_{3})||.\]  
where \(\mathbf{c}_{i}\) are cells coordinates and \(f\) is the mapping from Algorithm 1. In other words, the mapping function is not an increasing function of the well distance. It follows that if the kernel function utilizes the distances of the mapped input space directly, there will likely be a discrepancy between the calculated covariance and the actual covariance between inputs when this translates to the original input space. To overcome this problem, the kernel function was modified so that covariance is calculated based on the inverse mapping of the input space, i.e., from the \(N\)-dimensional input space to the \(2N\)-dimensional one which now represents the actual distance between two cells \((\mathbf{x},\mathbf{y})=k(||f^{-1}(\mathbf{x})-f^{-1}(\mathbf{y})||)\), where \(f^{-1}(\mathbf{x})=(f^{-1}(x_{1}),f^{-1}(x_{2}),\ldots f^{-1}(x_{11}))\). By utilizing this technique, the kernel function now considers the actual distance between two cells. The last issue to be addressed is the permutation invariance. Since each input vector is invariant under \(m!\cdot n!\) permutations, to calculate the representative covariance between two inputs, the permutation ordering that minimizes the distance of the one input to the  
Figure 5: Both ordered configurations produce the same output. other needs to be applied. This is a key modification that transforms BO from considering inputs as vectors to now view them as sets. By permuting, it is now able to minimize the distance between sets rather than vectors. To solve this, a matrix \(\mathbf{D}\) is generated, with each value \(d_{ij}\) representing the distance of the \(i\)th well in \(\mathbf{x}\) from the \(j\)th well in \(\mathbf{y}\), as given by Equation (16).