## 1 Introduction
This method involves studying the properties of the geomagnetic field and applying a simple yet robust mathematical model to train on geomagnetic datasets. The LSTM outperforms current methods by providing highly accurate predictions during both calm and disturbed geomagnetic conditions, with noticeably reduced average and maximum errors. Others use more general Artificial Neural Networks (ANNs) for magnetic source localization by using the Magnetic Gradient Tensor (MGT) [27]. [PERSON] et al. proposed a novel approach for magnetic target localization, leveraging an ANN alongside magnetic field intensity and MGT signals. The experimental findings confirmed the viability of the method, with an average localization error of 2.15 cm for targets positioned between 61 and 79 cm. In our study, we address the challenge of accurately mapping magnetic fields using a sparse 3 \(\times\) 3 array of PHE sensors. To overcome the inherent limitations of sparse sensor networks, we implement both deterministic and ML-based approaches for data processing. Specifically, we employ LMA to estimate the position and characteristics of the magnetic source and reconstruct the field map. While LMA provides a good solution for the localization problem, its accuracy is limited by its dependency on initial guesses for variables and its inability to resolve multiple sources effectively. To achieve better results than those achieved using the LMA, we solve the same problem with a different method. We introduce an FCNN [28] trained on simulated data, which significantly enhances spatial resolution and field mapping accuracy without requiring additional sensors. This combination of traditional and ML-based methodologies highlights the advantages and trade-offs in reconstructing magnetic fields using minimal hardware. The main contributions of our study can be summarized as follows:  
1. Reduction in Sensor Count: Traditional magnetic field mapping methods require arrays with more than 50 sensors to achieve high spatial resolution. This work demonstrates equivalent results using only nine sensors with the help of machine learning algorithms;
2. ML-Powered Mapping: An FCNN was developed to process data from the sparse sensor array, significantly enhancing the magnetic field map's resolution and accuracy, without relying on a specific physical model, compared to deterministic methods;
3. Use of PHE Sensors for Magnetic Field Mapping: This study is the first to employ PHE sensors in a compact \(3\times 3\) grid for precise magnetic field mapping, highlighting their potential for high-resolution applications;
4. Application of Small Sensors: The use of small elliptical PHE sensors demonstrates their feasibility for constructing dense sensor arrays in constrained environments, such as medical and industrial applications. The structure of this paper is as follows:  
Section 2 details the experimental setup, including the design of the PHE sensor array, the magnetic source, and the data acquisition system. This section also describes the deterministic and machine learning-based data processing methods used. Section 3 provides an in-depth analysis of the experimental and simulation results. This includes the performance of the LMA and the FCNN in reconstructing magnetic field maps, along with a comparison of their effectiveness. Section 4 concludes and explores the implications of the results, emphasizing the scalability and applicability of the proposed methods.