## 5 Informative Path Planning for Mobile Network Mapping  
The previous sections presented how physics-informed GPs can help create a reliable three-dimensional map of a cellular network site. The following elaborates on these results by defining a path for the UAV to collect measurements and create a reliable map with a fixed travel budget. The path should minimize the root-mean-square error between the ground-truth measurement data and the prediction of our physics-informed GP model. It should be adjusted as new measurements become available that allow for updating of the GP posterior. Such a path-planning problem is also known as informative path planning (IPP). The probabilistic nature of the GP-based model encourages the adoption of a Bayesian optimization (BO) approach, as detailed previously by [PERSON] et al. (2018). BO is a highly efficient optimization technique for functions that demand expensive evaluations. However, BO, as a global optimizer, suggests a single point (\(\mathbf{x}^{*}\)) to evaluate next to find the optimum function value. To iteratively define the UAV's trajectory for an IPP mapping task, a candidate selection approach similar to those proposed by [PERSON] et al. (2019) and [PERSON] et al. (2020) is used. For a specified number of random points (\(\mathbf{p}_{i}=\mathbf{x}_{i}^{*}\)) within the mapping area, a BO utility function considering both the mean and uncertainty of the GP is evaluated. Additionally, the Euclidean distance from the UAV's current location to each candidate point, denoted as \(d_{i}\), is computed. This approach is visualized in Figure 7. It is important to note that unlike the depicted points and the approaches proposed by [PERSON] et al. and [PERSON] et al., the candidate points are chosen in a three-dimensional search space instead of on a two-dimensional plane. Figure 7: Simplified 2D visualization of the candidate points. The BO utility value is calculated using an upper confidence bound (UCB, \(a_{\text{UCB}}\)) modified to identify regions with a high gradient (cf., [12]):  
\[\
abla a_{\text{UCB}}[\mathbf{p}_{i}]=|\
abla\mu_{\text{GP}}(\mathbf{p}_{i})|+ \kappa\sigma_{\text{GP}}(\mathbf{p}_{i}). \tag{8}\]  
where \(\
abla\mu_{\text{GP}}\) and \(\sigma_{\text{GP}}\) represent the gradient of the predictive mean and the variance of the GP, respectively. The \(\kappa\) parameter balances exploration and exploitation. In order to also assess data points that would be collected on the way to the candidate point, the BO utility is summed up along a straight line to the candidate point using a fixed interval that matches the flight speed (\(v\)) of the UAV:  
\[r(\mathbf{p}_{i};M_{i})=\frac{1}{M_{i}}\sum_{m=0}^{M_{i}}\
abla a_{\text{UCB}} [l(\mathbf{p}_{i};m,M_{i})]. \tag{9}\]  
with \(M_{i}=\lfloor d_{i}/v\rfloor\) and  
\[l(\mathbf{p}_{i},m,M_{i})=\mathbf{p}_{0}+\frac{m}{M_{i}}(\mathbf{p}_{i}- \mathbf{p}_{0}), \tag{10}\]  
where \(\mathbf{p}_{0}\) denotes the present location of the UAV. Both the BO reward and the distance to the candidate point are individually ranked, with the highest BO reward (\(r\)) and the smallest distance (\(d\)) receiving the highest rank. This normalization facilitates a direct trade-off achieved by summing the ranks with a tuning factor (\(\alpha\)), which provides flexibility to tune the path length to the next waypoint. Subsequently, the UAV navigates to the candidate point with the highest summed rank, acquiring new measurements along the way. In the subsequent iterations, the GP model is updated with the new data, and fresh candidate points are selected and evaluated. This iterative process continues until the summed-up path lengths reach a travel budget or a predefined maximum number of iterations is surpassed. Iterative path planning is summarized in Algorithm 1. ```
1\(k=1\)
2\(b\leftarrow\) travel budget
3\(maxIter\leftarrow\) max no.