# Features Leverage in Graph Models for Mineral Prospectivity Mapping

Thi Hai Yen Vu, Thi Bich Hanh Dao, Vincent Nguyen, Christel Vrain, Hugo Breuillard

► To cite this version:

Thi Hai Yen Vu, Thi Bich Hanh Dao, Vincent Nguyen, Christel Vrain, Hugo Breuillard. Features Leverage in Graph Models for Mineral Prospectivity Mapping. ACM Symposium On Applied Computing (SAC), Mar 2025, Sicily, Italy. pp.597-604, 10.1145/3672608.3707885. hal-04967663

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# Features Leverage in Graph Models for Mineral Prospectivity Mapping

Thi Hai Yen VU

thi-hai-yen.vu@univ-orleans.fr

Université d'Orléans, LIFO UR 4022

BRGM, F-45060 Orléans

Orléans, France

Thi Bich Hanh Dao

thi-bich-hanh dao@univ-orleans.fr

Université d'Orléans, LIFO UR 4022

Orléans, France

Vincent Nguyen

vincent.nguyen@univ-orleans.fr

Université d'Orléans, LIFO UR 4022

Orléans, France

Christel Vrain

christel.vrain@univ-orleans.fr

Université d'Orléans, LIFO UR 4022

Orléans, France

Hugo Breuillard

h.breuillard@brgm.fr

BRGM, F-45060 Orléans

Orléans, France

## Abstract

Mineral Prospectivity Mapping (MPM), the process of identifying areas with high potential for mineral deposits, can be divided into two main categories: knowledge-driven and data-driven. Knowledge-driven techniques rely on expert opinion on geological data, while data-driven techniques employ ML models to predict the probabilities of mineral occurrences based on known geological datasets. Recently, with the advancement of machine learning (ML) methods, data-driven MPM has gained significant improvements. Notably, graph-based approaches overcome the disadvantages of previously used approaches (pixel-based, image-based) and have demonstrated better performances. However, the graph construction in current methods is based solely on spatial distances between pixels, regardless of their geological attributes. In this paper, we introduce a novel graph construction approach that combines spatial distances with other distances obtained from feature mining. Our experiments show that this combination outperforms existing graphs, and can be considered as a promising approach to integrate feature mining into data-driven models in MPM.

## CCS Concepts

* Applied computing → Geographical Information Analysis.

### ACM Reference Format:

Thi Hai Yen VU, Thi Bich Hanh Dao, Vincent Nguyen, Christel Vrain, and Hugo Breuillard. 2025. Features Leverage in Graph Models for Mineral Prospectivity Mapping. In *The 40th ACM/SIGAPP Symposium on Applied Computing (SAC '25), March 31-April 4, 2025, Catania, Italy*. ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/3672608.3707885

## 1 Introduction

Mineral Prospectivity Mapping (MPM) is the process of creating a prospectivity map of mineralization over specific regions [19]. In general MPM, a map is usually divided into pixels, each corresponding to a region with a certain size in reality (see examples in Fig. 1). The MPM's target is to analyze the geological data and then predict

the favorability score of each pixel corresponding to the region's mineralization probability. The results showing the high potential of mineral occurrences will be utilized for mineral exploration.

Generally, MPM techniques can be divided into 2 main categories: knowledge-driven and data-driven. Knowledge-driven techniques rely on expert knowledge of geological data to find prospective regions of minerals. Most of knowledge-driven methods such as boolean logic [4] and multi-criteria decision analysis [3] have a tendency to use subjective judgment based on experts' opinions for selecting criteria and customizing weights for each data feature. On the other hand, data-driven techniques employ machine learning (ML) models to predict the probabilities of mineral occurrences based on known geological datasets. Previous data-driven MPM approaches usually consider location in the map as one independent data point and predict the mineral prospectivity given the data features using ML methods such as Random Forest [1], SVM [18], and Neural networks [16]. Recently, new approaches have been developed that take into account the spatial relationship between the locations, including image-based [10] and graph-based approaches [28]. Notably, graph-based approaches [27, 28, 35] have gained more attention and have been demonstrated to perform better in multiple scenarios over the other approaches. In graph-based approaches, the map is considered as a large graph, where each pixel is regarded as a node of the graph. The edges are constructed based on the spatial distances of the pixels. However, this approach disregards the geological information contained in the pixels in the graph construction. Hence, the graph could incorrectly associate nodes that lack homophily with respect to critical geological attributes. That may result in ineffective connections between nodes.

In this work, we propose a new method that integrates geological features in the graph construction phase of graph-based MPM approach by using a geological attribute distance in addition to the spatial distance between pixels. We applied the Jaccard distance to measure the dissimilarity between the geological formations which are considered as attribute distances. The geological attribute distances obtained from formation and fault features are combined with spatial distances to identify neighboring nodes and construct graph edges. To evaluate our approach, we perform experiments to investigate antimony (Sb) occurrences in the Armorican Massif region of France. We show that our method outperforms classic ML-based methods as well as current graph-based methods on our

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This work is licensed under a Creative Commons Attribution 4.0 International License.

SAC '25, March 31-April 4, 2025, Catania, Italy

© 2025 Copyright held by the owner/authors.

ACM ISBN 979-8-4007-0629-5/25/03

https://doi.org/10.1145/3672608.3707885
dataset. Our work introduces a new approach to integrate feature mining into graph-based approaches, opening up new possibilities for applying dataset knowledge to data-driven models in MPM.

The paper is organized as follows. Section 2 presents some preliminary notions including the general MPM problem setting and the graph-based approach. Section 3 discusses related work including different data-driven MPM approaches. In Section 4, we describe the motivation of our work, the dataset as well as our proposed method. Next, we present our experiment setup in Section 5, the results are given in Section 6. Finally, we further discuss our method in Section 6.4 and talk about the perspectives in Section 7.

# 2 Preliminaries

## 2.1 MPM problem setting

Generally, MPM data is raster data represented by a (r × c) grid (map) of r rows and c columns. Each grid square is referred to as a pixel. Within a specific region of interest, a subset of n pixels (n ≤ r × c): X = {X₁, X₂, ..., Xₙ} is considered, where each pixel Xᵢ has the corresponding coordinate (rᵢ, cᵢ) on the grid. Each Xᵢ can be considered as a vector: Xᵢ = (xᵢ¹, xᵢ², ..., xᵢᵈ) ∈ ℝᵈ, where xᵢⱼ corresponds to a geological feature (e.g. fault, rock types).

The set of the corresponding mineral deposits is denoted as Y = {y₁, y₂, ..., yₙ}, where:

$$y_i = \begin{cases} 1 & \text{if there is a mineral deposit in } (r_i, c_i) \\ 0 & \text{if there is no mineral deposit in } (r_i, c_i) \\ -1 & \text{(unknown) if there is no information} \end{cases}$$

Each pixel Xᵢ is called positive if yᵢ = 1, negative if yᵢ = 0 and unlabeled if yᵢ = -1. Since ore-forming is a difficult process requiring a lot of hard condition, the known mineral occurrences is pretty rare in reality. Therefore, the number of positive examples Xᵢ is very small (|{yᵢ = 1}| << |Y|). With the high cost for mining exploration, almost data in MPM is unlabeled (|{yᵢ|yᵢ = unknown}| ≈ |Y|). The objective of MPM is to predict the mineral prospectivity of the unlabeled pixels.

## 2.2 Graph-based MPM

Graph-based MPM approaches consider each pixel as a node of a graph and turns the MPM problem to a node classification problem. Formally, the MPM data is constructed as an unidirected graph G = (V, E) where V is the set of n vertices V = {V₁, V₂, ..., Vₙ} and E is the set of edges. Each vertex Vᵢ has the corresponding feature vector Xᵢ ∈ ℝᵈ, and the mineral deposits yᵢ are considered as the targets for the model. In existing graph-based MPM approaches [27, 35], the edges E are defined by the spatial distance of the actual pixels in the grid map (e.g. Vᵢ and Vⱼ are neighbors if max(|rᵢ - rⱼ|, |cᵢ - cⱼ|) ≤ 1, where: (rᵢ, cᵢ), (rⱼ, cⱼ) are corresponding coordinations of Xᵢ, Xⱼ). The constructed graph is trained with the node classification task using Graph Neural Networks (GNN).

The main idea of GNN models is to generate a node representation by aggregating its own representation and its neighbors' representation in each layer. In detail, the representation h_v of a node v ∈ V is updated in the layer l + 1:

$$h_v^{(l+1)} = \sigma \left( \sum_{u \in N(v) \cup \{v\}} \alpha_{uv} \times W^{(l+1)} \times h_u^{(l)} \right) \quad (1)$$

where N(v) is the set of all neighbors of v, α_{uv} is the coefficient measuring the connective strength between 2 nodes u and v, h_v^{(l+1)} is the node representation of v in the layer l + 1 (h_v^0 = x_v), W^{(l)} ∈ ℝ^{F^{(l)}×F^{(l+1)}} is shared weight matrix in the l-th layer (F^{(l+1)}: dimension of node representation at layer l + 1) and σ is the activation function ReLu(x) = max(0, x).

Despite sharing the same mechanism to update the node's representation, each graph network architecture uses a different way to calculate the coefficient α_{uv}. Two primary network architectures that are widely used are: Graph Convolution Network (GCN) [9] and Graph Attention Network (GAT) [21]. In GCN model, the coefficient α_{uv} is constant and the same for all layers:

$$\alpha_{uv} = \frac{1}{\sqrt{\text{deg}(u) \times \text{deg}(v)}} \quad (2)$$

where deg(u), deg(v) are the degree (number of neighbors) of nodes u, v respectively. GAT model introduced the attention mechanism where the α_{uv} changes in each layer and is a trainable parameter:

$$\alpha_{uv}^{(l+1)} = \sigma \left( a^T [W^{(l+1)} \times h_u^{(l)} || W^{(l+1)} \times h_v^{(l)}] \right) \quad (3)$$

where || is the concatenate operation of 2 vectors and a ∈ ℝ²F^{(l+1)} is a weight vector to parameterize the attention mechanism, W^{(l)} ∈ ℝ^{F^{(l)}×F^{(l+1)}} is shared weight matrix in the l-th layer. In summary, the coefficient α_{uv} in GCN reduces the influence of highly connected nodes in the graph, while in the GAT mechanism, it enables the model to assign different levels of importance to neighboring nodes based on their features and relevance to the target node.

# 3 Related work

This section summarizes common ML-based MPM methods which can be categorized into 3 main approaches depending on how the data is processed: pixel-based, image-based, and graph-based approaches (see Fig. 1).

**Pixel-based approach.** This is one of the most classic approaches to MPM problems. In pixel-based approach, each pixel Xᵢ is considered as an independent sample with geological features (xᵢ¹, xᵢ², ..., xᵢᵈ) as inputs. The MPM is considered as a ML classification problem with tabular data. This approach is widely used when combining with different ML models. For instance, [18] employs an ensemble strategy that combines the predictions of three algorithms: Random Forest, Support Vector Machine and Maximum Likelihood to boost the performance. Similarly, [23] combined semi-supervised learning with RF to tackle the lack of labeled data in MPM. [20] proposed a Spatial Random Forest (SRF) technique which captures local spatial-spectral information from MPM data. To handle the imbalance of MPM positive and negative labels, [1] used Gradient Boosting Tree combined with Focal loss. Subsequently, [16] introduced Neural Networks (NN) to solve the MPM problem leading to the application of other deep learning architectures such as Gated Recurrent Unit [31], LSTM [24], and Deep Auto-Encoder [26]. Overall, these methods extract and integrate the deep-level geological information to illustrate the effectiveness of pixel-based approach
<figure><img src="image_4.png" /><figcaption>**Figure 1:** Overview: pixel-based, image-based, graph-based approaches.</figcaption></figure>

in tackling MPM problems. However, one limitation of the pixel-based approaches is their inability to capture spatial information, an important factor in predicting mineral deposits [33].

**Image-based approach.** The issue of ignoring spatial information in the pixel-based approach can be addressed by using an image-based approach. Instead of considering independent pixels, this approach consider images cropped from the initial map (see Figure 1). Each image has the predictive pixel $X_i$ in its center. If the center pixel is positive then the image is considered as positive and vice versa. The MPM problem then turns into an image classification problem. To classify images, this approach generally employs Convolutional Neural Networks (CNNs) with convolutional filtering and pooling mechanisms, which allow the model to learn not only from the individual pixels but also from the surrounding pixels. This leads to effectively capture spatial relationships within each image [34]. Some variants of CNN models are applied on MPM such as VNET [30], UNET [11], CNN - Transformer [2] to boost the performance of image classification. Besides, various techniques have been developed such as random-drop [10], direct sampling [25] for data augmentation to alleviate the issue of training sample scarcity in MPM image-based approaches. Despite demonstrating a stronger ability to capture spatial information, the image-based approach has certain limitations. It only considers spatial relationships between pixels within the image, missing out on adjacent pixels that may fall outside the image boundaries. Additionally, this approach relies on fixed-size images, which limits its ability to provide a favorability score of the boundary pixel in the map.

**Graph-based approach.** Recently, graph-based approaches were introduced to overcome the limitations of the two previous approaches. For this approach, a graph is constructed where each pixel is a node and the edges of the graph are built based on the spatial information of the pixels. Each pixel is connected to the surrounding pixels on the map (see Figure 1). The MPM problem then turns to a graph node classification. Among the papers using on this approach, [35] analyzed the advantages of graph-based approaches to capture spatial relationships of mineralization information. Besides, their experiments demonstrated that this approach

outperforms the image-based approaches in predicting areas where mineral deposits are found and showed that image- based models have tendency to predict too large an area with fewer actual deposits. Additionally, [27] used post-hoc explanation to show that graph-based approaches can identify more effectively important attributes in mineralization when compared to the others. However, all mentioned paper used graphs whose edges are constructed in a simple way based only on the spatial distance. In our work, we suggest a novel way for graph construction where the edges are chosen based on not only the spatial distance but also various geological attributes (e.g. fault, formation, magnetics, etc).

# 4 Methods

## 4.1 Motivation of our work

In node classification tasks, GNNs tend to perform well when graphs exhibit homophily [12], where nodes with the same label or, in cases of missing labels, nodes with similar features are more likely to be connected. In the MPM context, nodes with similar geological attributes should be considered neighbors, even if they are not geographically adjacent. Without considering geological feature distances, the graph might incorrectly connect nodes that do not exhibit homophily in terms of important geological features. This misalignment can lead to over-smoothing [29], where nodes with different geological attributes (e.g., positive and negative mineral deposits) become indistinguishable due to the similarity in their representations. We expect that incorporating feature distances into the neighbor selection process, will better align the graph structure with geological homophily to improve node classification results.

## 4.2 Proposed graph construction

4.2.1 *General MPM dataset.* In the MPM dataset, each feature $x_i$ corresponds to a geological feature. These features can be under the forms: Boolean/ binary (e.g. presence or absence of a formation), numeric (number of faults or the concentration of specific elements) or intervals (e.g. ranges of geochemical values).
Depending on the dataset, the set of d features of {$x^1, ..., x^d$} combined with spatial location can be divided into k different attributes, each of them can be a feature or a set of features with the idea that each attribute encapsulates a semantic information. Indeed, in Geographical Information Systems (GIS), information is usually divided into layers, an attribute could be associated to each layer, thus corresponding to different semantic data. Information in the layers can be encoded differently. For instance, a categorical feature could be one-hot encoded by a set of Boolean features representing this encoding, and the attribute would be this set of features. Therefore an attribute can be a feature (e.g. $x^k$: fault) or a set of different features ($m < d$) (for instance set of geological formation features or set of geochemical features).

### 4.2.2 Graph construction.
For the existing graph-based MPM approaches (mentioned in 2.2), the spatial distance is the only attribute to decide the neighbors. More precisely, $V_1$ and $V_2$ are neighbors if the spatial distance between $X_1$ and $X_2$ satisfies:

$$d^{spatial}(X_1, X_2) = \max(|r_i - r_j|, |c_i - c_j|) \le 1. \quad (4)$$

Rather than relying solely on spatial attributes for neighbor selection, we propose utilizing various geological attributes to better identify neighboring nodes. Based on the distances between these geological features, we will determine whether two corresponding nodes should be considered neighbors (Fig. 2). In order to better exploit information that is usually only presented in the nodes of the graph, we aim at taking into account of it when building the edges and we propose to use a weighted distance computed by:

$$d_{new}(X_i, X_j) = \sum_{p=1}^{k} \pi_p d^p(X_i, X_j) \quad (5)$$

where $d^1, d^2, ..., d^k$: distances of k corresponding attributes (e.g. spatial distance, fault distance, ...), $\pi_1, \pi_2, ..., \pi_k$: (hyperparameters) weights of corresponding distances.

If the distance between $X_i$ and $X_j$ is smaller than a threshold K, i.e. $d_{new}(X_i, X_j) < K$, then the 2 corresponding nodes $V_i$ and $V_j$ are considered as neighbors.

## 4.3 Application on our dataset

### 4.3.1 Dataset.
In this work, we investigate the occurrences of antimony (Sb) in the Armorican Massif, located in the Northwestern part of France. The dataset includes 3 different geological layers over the same region: Geological formation, Faults, and Occurrences of Antimony [5, 6]. Following the data preprocessing of [1], we present geographic data by dividing the region into a grid of small squares whose size is 2km, which is the optimal value shown by [22]. We finally obtain a synthetic map of lithological associations.

In brief, for each layer, the data preprocessing are followed below:

(1) **Geological formation layer:** To each pixel of the grid, we associate a feature giving all the geological formations that intersect with it. This feature is then one-hot-encoded, leading to as many Boolean features than the geological formation: each value is converted into a new categorical column and assigned a value of 1 (presence) or 0 (absence) to those columns.

(2) **Faults layer:** For each pixel, we associate a fault feature, giving the number of faults that intersect with itself.

(3) **Occurrences of Antimony:** To each pixel of the grid, we associate a feature whose value is 1 if there is at least 1 antimony deposit in the pixel and -1 otherwise. This feature is considered as the output of the model. There are no negative examples and the value -1 is given to pixels that are not known to contain deposits.

After this preprocessing, we define attributes: fault (1 feature), geological formation (the set of 117 features) and spatial information which will be considered for our graph construction.

### 4.3.2 Proposed distances for neighbor selection.
In our case, we investigate the potential occurrences of antimony, a rare metal having an average concentration of up to 0.4 ppm in the upper continental crust [17] and up to 0.8 ppm in the oceanic crust [8]. Research on antimony mineralization in the Armorican Massif has demonstrated that these hydrothermal deposits are usually spatially associated with regional-scale faults [13]. Besides, geological formations are considered as crucial to antimony occurrences due to the evidences including localisation of antimony quartz veins [13], the relationship between Sb mineralization and mafic magmatism [14] as well as mafic event [15]. Therefore, the above mentioned studies conclude that besides spatial location, faults and geological formations are important to the antimony mineralization in Armorican Massif.

We propose 3 different distances based on these 3 attributes: spatial location, fault presence and group of geological formations.

(1) **Spatial distance:** For 2 pixels $X_i$ and $X_j$ with the corresponding coordinate $(r_i, c_i)$, $(r_j, c_j)$, the spatial distance is:

$$S(X_i, X_j) = d^{spatial}(X_i, X_j) = \max(|r_i - r_j|, |c_i - c_j|) \quad (6)$$

(2) **Fault distance:** For 2 pixels $X_i$ and $X_j$ with the corresponding fault feature $x_i^{fault}, x_j^{fault}$, the fault distance is:

$$F(X_i, X_j) = d^{fault}(X_i, X_j) = |x_i^{fault} - x_j^{fault}| \quad (7)$$

(3) **Geological formation distance:** In our dataset we have 117 geological formation features, which are one-hot encoded (value of 1 if presence and 0 if absence), and we use the Jaccard distance [7] to calculate the geological formation distance (GFD). The GFD of 2 pixels $X_i$ and $X_j$, which estimates the dissimilarity of geological formations, is:

$$P(X_i, X_j) = d^{formation}(X_i, X_j) = 1 - \frac{|s(X_i) \cap s(X_j)|}{|s(X_i) \cup s(X_j)|} \quad (8)$$

where $s(X_i)$ and $s(X_j)$ are the corresponding subsets of formation attributes of $X_i$ and $X_j$ such that: $s(X_i) = \{k | x_i^k = 1\}$, $s(X_j) = \{k | x_j^k = 1\}$, where the $x_k$ represent the one-hot encoding of the geological formation.

The condition to choose neighbors can therefore be written as: $X_i$ and $X_j$ are neighbors if

$$\pi_{spatial}S(X_i, X_j) + \pi_{formation}P(X_i, X_j) + \pi_{fault}F(X_i, X_j) \le K \quad (9)$$

where the weights $\pi_{spatial}, \pi_{formation}, \pi_{fault}$ and the threshold K are some hyperparameters to be chosen. Different choices of hyperparameters may have different impacts on the performance
<figure><img src="image_5.png" /><figcaption>Figure 2: Existing graph construction: Neighbors chosen based only on spatial distance (that means if 2 pixels are adjacent then 2 corresponding nodes are neighbors). Our proposed methods: Besides the spatial distance, if the geological attributes of 2 pixels have (nearly) the same characteristics then 2 corresponding nodes are considered neighbors.</figcaption></figure>

of the model. We show in Section 6.2 the effects of different hyper-parameters on model performances.

## 4.4 Proposed method

The MPM dataset has an important issue, it contains very few positive samples and a lot of unlabeled data. To address this issue, two main methods are usually used to consider negative samples for training a binary classification model. The first method consists of considering all unlabeled samples as negative [1]. However, based on our experiments, the significant imbalance between 2 classes leads to over-smoothing in graph neural networks, where the model incorrectly classifies all nodes as negative. The second method involves randomly selecting negative samples from the unlabeled set [35]. We adopted this method and selected from unlabeled data a random subset of negative samples whose cardinality is the same as the positive ones to ensure a balanced training data. Nevertheless, we observed biases in the model towards the selected negative samples. Training on a small, randomly chosen subset causes overfitting, making it difficult for the model to classify new samples that differ from the selected negatives.

To overcome this, we propose a method inspired from ensemble technique to mitigate bias and improve performance. Our approach involves training *k* classifiers, each on a different cross-validation set, meaning that each model is exposed to a different subset of negative samples. We then average the predictions, resulting in a more balanced final output that is less influenced by any single negative subset. While individual models may develop bias toward their specific negative sets, the ensemble method helps mitigate this issue. Details of the data setup for training and the testing procedure will be provided in Section 5.

# 5 Experiment setup

## 5.1 Data setup and baselines

5.1.1 Data setup. Following the data preprocessing explained in Section 4.3.1, the dataset contains *n* = 14218 pixels, each of which has *d* = 118 features, including 117 geological formation binary features (0/1) and 1 fault feature. There are 105 positive samples (labeled as 1) and the rest is unlabeled. We create 10 cross-validation independent experiments. For each experiment, we process the data for the ensemble strategy as follows:

* Eleven positive samples (10% of all positive samples) were randomly chosen as the test set.

* From the remaining samples, we created 10 cross-validation sets from the 94 positive samples left, each containing 85

positive samples for training and 9 positive samples for validation. For training the binary classifier, we use the 85 positive samples and we randomly sample 85 negative samples from the set of unlabeled data as discussed in Section 4.4. This results in a total of 10 classifiers for each experiment. The averaged predictions from these 10 classifiers are then used for the final evaluation on the test set.

To ensure a fair comparison between different methods, we keep the same data setup (test set, cross-validation sets and negative samples) across all methods in each experiment.

5.1.2 Baselines. We compare our method with two baselines:

(1) **Pixel-based Approaches.** We compare our method to two pixel-based approaches, which do not consider the spatial attribute [16, 32]: XGBoost ensemble learning and multi-layer perceptron (MLP) models.

(2) **Graph-based Approach.** We conducted experiments on the baseline graph construction method based only on the spatial locations on the map [35] (mentioned as defined in Equation 4), which we denoted as $G_0$.

## 5.2 Evaluation

**Recall at Top k% Metrics.** For binary classification problem, popular metric choices include accuracy or precision. However, in MPM, there are usually very few positive samples compared with unlabeled samples. In our case, we have 105 positive samples among the total 14218 samples, with no negative sample. Therefore, we employ another metric to evaluate the models based on only known positive samples. This metric is derived from the recall score but here, we consider only top *k%* samples having the highest probability predictions instead of all samples predicted positive as in [35]. In detail, assume that the probability predictions given by the method over the *n* pixels of the dataset *X* are sorted such that $p_{model}(X_{s1}) \ge p_{model}(X_{s2}) \ge \dots \ge p_{model}(X_{sn})$, then the recall at top *k%* metrics on the testset $P = \{X_{p1}, \dots, X_{pm}\}$ of *m* positive samples is calculated as:

$$R_k(P) = \frac{|\{p_1, \dots, p_m\} \cap \{s_1, \dots, s_l\}|}{m} \quad (10)$$

where $l = \lfloor n \times k/100 \rfloor$ and $\{s_1, \dots, s_l\}$ denotes the top *k%* of model predictions, the numerator denotes the number of positive samples in the top *k%* of model predictions, and the denominator denotes the number of true positive samples.

**Method Evaluation:** For each method evaluation with a specific set of hyperparameters, we conduct 10 independent experiments. In each experiment, we train 10 single models and generate final
predictions using the ensemble approach mentioned in Section 4.4 that averages the predictions of these models. This results in two types of outcomes for each experiment: (1) the mean score of the 10 single models corresponding to 10 cross-validation sets for each experiment mentioned in Section 5.1.1, and (2) the mean score from the ensemble strategy of 10 experiments. To analyse the performance of the single models and the ensemble in Table 4, we compute the average of both (1) and (2), providing the overall performance of the single model and the ensemble strategy. Unless otherwise specified, the scores reported in Table 2 and Table 3 in Section 6 are the performance of the ensemble strategy for all methods (including baselines).

## 5.3 Implementation details

**Model.** For the baselines mentioned in Section 5.1.2, we design different implementation settings. For pixel-based approaches, we apply two different ML frameworks: XGBoost model (Python library with early stopping rounds=10) and Neural Network (2-layer MLP with 16 hidden features, implemented in pytorch). Besides, for graph-based approaches, we used common graph neural networks, namely GCN [9] and GAT [21]. We train the graph models in the transductive learning setting, where train and test embedding nodes comes from the same input graph but only test labels are used. Both models have 2 layers with hidden dimension 16. We use the only 1 attention head for GAT model. The graph models are implemented using torchgeometric. For our graph construction method, we use the same GCN and GAT models as the graph baseline approach.

**Optimization.** For the neural network models (MLP, GAT, GCN), we employ the Adam optimizer with learning rate 0.01, weight decay 5e-4 and the Cross Entropy Loss.

**Hyper-parameter selection.** The selection of hyper-parameters is a crucial step to see the influence of our method to graph-based approach in MPM. We performed a grid-search strategy to find the best weights $\pi_{spatial}$, $\pi_{formation}$, $\pi_{fault}$ and threshold K for defining neighbor selection as given in Equation 9. Without loss of generality, we can consider $\pi_{spatial} = 1$. The neighbor selection condition can then be written as: $S + \pi_{formation}^P + \pi_{fault}^F \le K$. The search space of each hyperparameter ($\pi_{formation}$, $\pi_{fault}$, K) is detailed in Table 1.

<table><caption>**Table 1: Hyperparameter search space for the neighbor selection condition $S + \pi_{formation}^P + \pi_{fault}^F \le K$**</caption><thead><tr><th>Hyperparameter</th><th>Considered Values</th></tr></thead><tbody><tr><td>π<sub>formation</sub></td><td>{0.5, 1.0, 1.5, 2.0}</td></tr><tr><td>π<sub>fault</sub></td><td>{0.2, 0.3, 0.5, 1.0}</td></tr><tr><td>K</td><td>{1.0, 2.0, 3.0, 4.0}</td></tr></tbody></table>

The selection criterion is determined for our experiments using recall at top 5% (mentioned in Section 5.2) on the validation sets obtained after running experiments on the data setup in Section 5.1.1. Selected values of all hyperparameters are: $\pi_{formation} = 1.0$, $\pi_{fault} = 0.2$, K = 4, that means, after grid search strategies, the graph $G_1$ is constructed under the neighbor selection condition:

$$S + P + 0.2F \le 4. \qquad (11)$$

# 6 Results

This section shows the results of our method on the Armorican Massif dataset described in Section 4.3.1 using evaluation method described in Section 5.2.

## 6.1 Model performance

From Table 2, we first observe that graph-based approaches achieve superior results than pixel-based approaches, which indicates that MPM problem can benefit from applying the graph structure rather than considering the pixels independently.

<table><caption>**Table 2: Recall at top 5%, 10%, 20% of GCN and GAT models on baseline graph $G_0$ (based only on spatial location in Equation 4), $G_1$ (graph followed our methods with optimal hyperparameters in Equation 11) compared with XGBoost and MLP models (pixel-based approaches in Section 5.1.2).**</caption><thead><tr><th rowspan="2"></th><th rowspan="2">MLP</th><th rowspan="2">XGBoost</th><th colspan="2">GCN</th><th colspan="2">GAT</th></tr><tr><th>G<sub>0</sub></th><th>G<sub>1</sub></th><th>G<sub>0</sub></th><th>G<sub>1</sub></th></tr></thead><tbody><tr><th>R<sub>5</sub></th><td>21.6</td><td>22.9</td><td>40.7</td><td><strong>45.8</strong></td><td>33.2</td><td><u>36.2</u></td></tr><tr><th>R<sub>10</sub></th><td>39.0</td><td>33.5</td><td>59.8</td><td><strong>60.8</strong></td><td>46.5</td><td><u>50.3</u></td></tr><tr><th>R<sub>20</sub></th><td>54.5</td><td>54.4</td><td>74.1</td><td><strong>76.1</strong></td><td>70.4</td><td><u>70.5</u></td></tr></tbody></table>

Besides, we can see that our proposed graph $G_1$ outperforms the baseline graph $G_0$ consistently in recall at top 5%, 10%, 20% metrics. Specifically, our method achieves a 12.5% improvement on recall at top 5% metrics for GCN model (45.8 vs 40.7) and a 9% improvement for GAT model (36.2 vs 33.2). These results indicate that the integration of geological information into graph construction as proposed can enhance the performance of graph-based models in identifying regions with high mineral deposit potential.

Additionally, we can see that GCN model shows better results than GAT on both baseline graph $G_0$ and graph $G_1$ constructed using spatial and geological attribute distances.

## 6.2 Impact of hyperparameters

This section presents the results obtained when varying the hyper-parameters indicated in Section 4.3.2. We also perform experiments where the graphs are constructed using spatial distance combined with either fault distance, geological formation distance, or a combination of all three factors.

### 6.2.1 Spatial distance combined with geological formation distance.
From Table 3a, we see that doubling the weights of geological formation distance $\pi_{formation}$ (from 1 to 2) does not show improvements in low threshold, but is beneficial for higher thresholds. We also observe that the models tend to perform better when increasing the average degree. In general, the best observed conditions are $S + P \le 4$ and $S + 2P \le 4$.

### 6.2.2 Spatial distance combined with fault distance.
From Table 3b, we observe that decreasing the weight of fault distance $\pi_{fault}$ (from 1 to 0.2) generally leads to higher scores in both models. Unlike previous section, higher average degree doesn't result in better performances, with the condition $S + 0.2 \times F \le 2$ with an average degree of 10.7 giving the best results.
<figure><img src="image_6.png" /><figcaption>Figure 3: Antimony mineralization favorability maps of the Armorican Massif using pixel-based approach (MLP model) and graph-based approach (GCN model). Each pixel is colored according to its predicted mineralization probability, indicated in the colormap shown on the right panel. The maps 3a is generated by MLP model, 3b and 3c are generated by using GCN model on baseline graph G₀ and our proposed graph G₁ as proposed respectively. The known occurrences are marked by white stars.</figcaption></figure>

<caption>Table 3: Recall at top 5%, 10%, 20% of GCN and GAT models when the graphs are constructed with different neighbor selection conditions based on spatial distance S (Equation 6), geological formation distance P (Equation 8) and fault distance F (in Equation 7). Avg. degree is the average of node degrees in the graph.</caption>

(a) Spatial and geological formation distances

<table><thead><tr><th rowspan="2">Conditions</th><th rowspan="2">Avg. degree</th><th colspan="3">GCN</th><th colspan="3">GAT</th></tr><tr><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th></tr></thead><tbody><tr><td>S + P ≤ 2</td><td>8.6</td><td>38.9</td><td>58.1</td><td>76.1</td><td>32.3</td><td>41.7</td><td>69.5</td></tr><tr><td>S + P ≤ 3</td><td>23.8</td><td><u>42.8</u></td><td><u>62.9</u></td><td>76.2</td><td>37.1</td><td>50.3</td><td>67.7</td></tr><tr><td>S + P ≤ 4</td><td>46.3</td><td><u>43.9</u></td><td><u>62.9</u></td><td><u>76.3</u></td><td><u>38.9</u></td><td><u>51.3</u></td><td><u>76.3</u></td></tr><tr><td>S + 2P ≤ 2</td><td>6.7</td><td>34.1</td><td>51.1</td><td>69.7</td><td>29.5</td><td>38.0</td><td>62.7</td></tr><tr><td>S + 2P ≤ 3</td><td>16.5</td><td>38.2</td><td>56.0</td><td>74.2</td><td>36.0</td><td>45.7</td><td><u>72.3</u></td></tr><tr><td>S + 2P ≤ 4</td><td>33.0</td><td>41.8</td><td><u>59.2</u></td><td><u>77.1</u></td><td><u>39.8</u></td><td><u>52.2</u></td><td>69.5</td></tr></tbody></table>

(b) Spatial and fault distances

<table><thead><tr><th rowspan="2">Conditions</th><th rowspan="2">Avg. degree</th><th colspan="3">GCN</th><th colspan="3">GAT</th></tr><tr><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th></tr></thead><tbody><tr><td>S + 0.2F ≤ 2</td><td>10.7</td><td><u>37.8</u></td><td><u>51.2</u></td><td><u>71.4</u></td><td><u>37.1</u></td><td><u>50.2</u></td><td><u>72.5</u></td></tr><tr><td>S + 0.3F ≤ 2</td><td>9.8</td><td>35.1</td><td>49.2</td><td>68.4</td><td>32.3</td><td>46.5</td><td>67.5</td></tr><tr><td>S + 0.5F ≤ 2</td><td>9.1</td><td>32.5</td><td>45.5</td><td>68.5</td><td>33.3</td><td>43.7</td><td>64.8</td></tr><tr><td>S + F ≤ 2</td><td>8.2</td><td>31.5</td><td>46.5</td><td>65.7</td><td>28.5</td><td>43.8</td><td>62.7</td></tr><tr><td>S + F ≤ 3</td><td>17.6</td><td>31.4</td><td>51.2</td><td><u>70.2</u></td><td>26.7</td><td><u>51.4</u></td><td><u>71.5</u></td></tr><tr><td>S + F ≤ 4</td><td>31.3</td><td><u>37.1</u></td><td><u>53.9</u></td><td>69.5</td><td><u>34.4</u></td><td>47.5</td><td>68.6</td></tr></tbody></table>

(c) Spatial, geological formation and fault distances

<table><thead><tr><th rowspan="2">Conditions</th><th rowspan="2">Avg. degree</th><th colspan="3">GCN</th><th colspan="3">GAT</th></tr><tr><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th><th>R<sub>5</sub></th><th>R<sub>10</sub></th><th>R<sub>20</sub></th></tr></thead><tbody><tr><td>S + 2P + F ≤ 3</td><td>6.7</td><td>28.8</td><td>44.7</td><td>62.7</td><td>28.5</td><td>38.0</td><td>61.8</td></tr><tr><td>S + P + F ≤ 3</td><td>8.8</td><td>29.7</td><td>48.5</td><td>67.7</td><td>30.3</td><td>41.9</td><td>65.7</td></tr><tr><td>S + P + 0.5F ≤ 3</td><td>11.5</td><td>36.1</td><td>51.4</td><td>68.5</td><td>31.5</td><td>46.5</td><td>64.8</td></tr><tr><td>S + P + F ≤ 4</td><td>18.2</td><td>34.3</td><td>49.3</td><td>69.4</td><td>33.4</td><td><u>54.3</u></td><td>69.5</td></tr><tr><td>S + 2P + 0.2F ≤ 4</td><td>22.8</td><td><u>42.8</u></td><td><u>64.7</u></td><td><u>74.0</u></td><td><u>34.1</u></td><td><u>50.3</u></td><td><u>74.3</u></td></tr><tr><td>S + P + 0.2F ≤ 4</td><td>31.8</td><td><u>45.8</u></td><td>60.8</td><td><u>76.1</u></td><td><u>36.2</u></td><td><u>50.3</u></td><td><u>70.5</u></td></tr></tbody></table>

6.2.3 Spatial distance combined with both fault distance and formation distance. From Table 3c, we also observe that better scores are

obtained with high geological formation distance $\pi_{formation}$, and low fault distance weight $\pi_{fault}$. We see that the best condition is not necessary the one with highest degree both $S + 2P + 0.2 \times F \le 2$ (22.8 average degree) and $S + P + 0.2 \times F \le 4$ (31.8 average degree) show equally good results.

## 6.3 Impact of ensemble technique

From Table 4, it is observed that the ensemble technique outperforms the individual model in both $G_0$ and $G_1$. For instance, the ensemble method achieves scores of 40.7 and 45.8 on the recall at top 5% metrics for $G_0$ and $G_1$ respectively, compared to 22.1 and 22.2 for the individual model. Besides, we see that $G_1$ we proposed outperforms $G_0$ on both individual and ensemble methods.

<table><caption>Table 4: Recall at top 5%, 10%, 20% of individual and ensemble methods for GCN model on baseline graph $G_0$ defined in Equation 4 and proposed graph $G_1$ defined in Equation 11.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Individual (GCN)</th><th colspan="2">Ensemble (GCN)</th></tr><tr><th>G<sub>0</sub></th><th>G<sub>1</sub></th><th>G<sub>0</sub></th><th>G<sub>1</sub></th></tr></thead><tbody><tr><th>R<sub>5</sub></th><td>22.1 ± 4.7</td><td><u>22.2 ± 4.2</u></td><td>40.7 ± 5.1</td><td><u>45.8 ± 4.2</u></td></tr><tr><th>R<sub>10</sub></th><td>39.8 ± 6.4</td><td><u>41.8 ± 6.0</u></td><td>59.8 ± 2.9</td><td><u>60.8 ± 4.7</u></td></tr><tr><th>R<sub>20</sub></th><td>61.8 ± 6.0</td><td><u>64.8 ± 7.8</u></td><td>74.1 ± 3.6</td><td><u>76.1 ± 5.9</u></td></tr></tbody></table>

## 6.4 Discussion

Figure 3 shows antimony mineralization favorability maps of the Armorican Massif using pixel-based MLP model (3a), GCN model with baseline graph $G_0$ (3b) and the graph $G_1$ obtained with our proposed method (3c). We first see that the maps 3b and 3c generated by graph-based approaches have smoother predictive regions than the MLP map, which is reasonable since pixel-based approaches don't consider the spatial information of the pixels, which is one of critical factors in predicting mineral occurrences.

Considering the graphs $G_0$ and $G_1$, we observe that the predictive map derived from the feature mining-integrated graph $G_1$ is
less smooth and less dependent on the spatial information of the pixels, as contrasted to the baseline graph G₀ where the predictive regions are too smooth and can span over large areas. This indicates that our approach mitigates the risk of over-smoothing and enhances graph model's ability to reflect the geological heterogeneity, which is consistent with the motivation our work described in Section 4.1. It is also observed that the baseline graph G₀ tends to demonstrate over-confidence in its predictions, where the predictions obtained from our proposed graph G₁ are less confident, which aligns with the reality that the antimony occurrences (and mineral occurrences in general) are very rare due to the challenging nature of the mineralization process.

# 7 Conclusion and perspective

In this work, we introduce a novel graph construction method in graph-based MPM approach that is based not only on spatial location in the map but also on geological attributes. The idea comes from the fact that geological information play an important factor in deciding the node's neighbors. We show that our method outperforms traditional pixel-based approaches and baseline graph-based approaches. The results indicate the potential of integrating knowledge in data-driven MPM approaches. In future work, we could further study on how to integrate expert's opinions on the importance of each attribute. We would also expect that our approach would be more efficient on other informative datasets as well as specific geological information. Our work also suggests a novel graph learning approach that focuses on the construction phase of the graphs based on feature mining. We hope to extend our work to other graph problems in the future.

# Acknowledgments

This work is funded by the project ANR-20-THIA-0017-01: AI.iO Artificial Intelligence in Orléans: Learning from heterogeneous data and expert knowledge. Applications in geological and environmental sciences.

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