## 4 Conclusions  
This study addresses the challenge of precise magnetic field mapping with minimal hardware, leveraging a sparse array of PHE sensors. By integrating traditional optimization techniques with machine learning, the research provides an innovative approach to enhance spatial resolution without increasing the sensor density. We introduce a novel method for quantifying the magnetic field using PHE sensors, as well as conducting measurements with numerous sensors (an array of sensors) to generate a magnetic heat map. In order to further improve the resolution of our magnetic map, we developed dedicated machine learning models that use simulated measurements from an array of sensors to learn the theoretical magnetic heat map of the specific unit under test and the distortion of the field that it generates. The findings reported in this research demonstrate that employing ML yields equivalent resolution for a measured magnetic heat map compared to the one obtained using a significantly larger number of sensors. The FCNN outperforms deterministic methods in accuracy and flexibility, achieving resolutions comparable to a much denser sensor arrays. The findings suggest that advanced ML models can effectively reconstruct magnetic fields, making them a promising tool for applications requiring minimal hardware. However, while the results show clear improvements in spatial resolution and efficiency, there are inherent trade-offs associated with this approach. The reliance on simulated data for neural network training introduces a dependency on the accuracy and representativeness of the simulations. Any deviations between the simulated and experimental conditions may lead to inaccuracies in the heat map generation. Additionally, the sparsity of the sensor array, while innovative, inherently limits the granularity of the raw data, requiring robust interpolation methods to reconstruct the magnetic field. This places significant demand on the ML model's ability to generalize and accurately infer details that are not explicitly captured by sensor measurements. Future work could focus on validating the robustness of the ML approach using diverse experimental conditions and investigating hybrid strategies that combine ML with additional sparse measurements to balance accuracy and resource efficiency. In future work, our objective is to enhance the quantitative accuracy of the magnetic heat map while also achieving a precise estimation of the magnetic object's location. This could have several applications, such as identifying and differentiating multiple magnetic objects on a map, as well as determining the geometric characteristics of the anomalies. A possible solution could, for instance, integrate the 'pix2 pix' method, as described in the study by [PERSON] et al. [37]. A significant challenge lies in effectively training the model due to the requirement of a large and diverse dataset. Other future work includes optimizing the ratio of space size to sensor count to balance accuracy and cost-efficiency; adapting the method for varied magnetic object geometries, dimensions, and materials to ensure versatility; and exploring advanced ML techniques such as the Super Resolution Generative Adversarial Network (SRGAN) [38] to further refine field reconstructions. M.V. : Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Hardware (equal); Validation (equal); Visualization (equal); Writing--original draft (equal); Writing--review and editing (equal). D.L. : Formal analysis (supporting); Investigation (supporting); Software (supporting); Validation (supporting); Writing--review. M.S. : Formal analysis (supporting); Investigation (supporting); Software (supporting); Validation (supporting); Writing--review and editing (supporting). A.G.: Formal analysis (supporting); Investigation (supporting); Writing--review and editing (supporting). R.A.: Conceptualization (supporting); Validation (supporting); Writing--review and editing (supporting); Supervision (equal). L.K.