GeoCFM: Positive-Only Conditional Flow Matching for Mineral Occurrence Sampling

Conference: ECCV 2026
Paper: ECCV Official
Area: Medical Imaging (preset category; actually Earth Science / Generative Models)
Keywords: Conditional Flow Matching, Positive-Only Learning, Mineral Prospectivity Mapping, Spatial Point Sampling, Epistemic Uncertainty

TL;DR

Addressing the absence of reliable negatives and the epistemic non-identifiability of unobserved subsurface geology in mineral exploration, GeoCFM recasts mineral targeting from deterministic score-map prediction into conditional flow matching over continuous spatial point densities, generating multi-hypothesis occurrence distributions without pseudo-negatives.

Background & Motivation

Critical mineral exploration is decision-making conducted under extreme uncertainty. In practical mineral prospectivity mapping, exploration teams integrate multi-source remote sensing and geophysical survey layers—such as aeromagnetic grids, gravity anomalies, geochemical assays, and structural lineaments—to prioritize spatial drilling targets. Over the past decade, however, the global discovery rate of major economic mineral deposits has steadily declined, while deep subsurface targets remain elusive. In this high-stakes setting, physical drilling is economically prohibitive, environmentally damaging, and irreversible.

From a computer vision and machine learning perspective, mineral prospectivity deviates sharply from standard supervised dense prediction or binary classification due to two fundamental structural challenges. First, ground-truth supervision is inherently positive-only (PO) or positive-unlabeled (PU): verified mines and mineral occurrences are spatially sparse and exhibit severe historical exploration bias, whereas unlabeled regions simply represent unexplored, untested, or presently uneconomic ground rather than verified barren negatives. Conventional data-driven prospectivity pipelines fabricate pseudo-negatives by uniformly sampling background pixels, which systematically distorts the learned decision boundaries and induces false overconfidence. Second, mineralization is governed by hidden geological variables—such as deep structural conduits, thermal fluid pathways, and hydrothermal alteration histories—that cannot be directly observed from surface or airborne surveys. This non-identifiability makes the inverse mapping from observations to deposit locations inherently one-to-many, dominated by severe epistemic uncertainty. Deterministic per-pixel score maps cannot represent multi-modal distributions and collapse multiple plausible subsurface outcomes into a single over-smoothed map.

The angle of attack in this work is to discard the pseudo-negative classification paradigm altogether and model mineral discovery as conditional spatial density estimation and generative point sampling under positive-only supervision. Core idea: formulate mineral prospectivity as generating continuous 2D spatial occurrence point sets conditioned on multi-channel geo-images, utilizing conditional flow matching (GeoCFM) guided by UNet raster features to transport Gaussian base noise to deposit coordinates without pseudo-negatives while explicitly capturing epistemic uncertainty.

Method

Overall Architecture

GeoCFM takes a multi-channel geo-image tensor \(d \in \mathbb{R}^{C \times H \times W}\) as conditioning input to model the conditional spatial probability distribution \(\pi(p \mid d)\) of sparse mineral occurrence coordinates \(p \in \mathbb{R}^2\). Because the underlying geological context \(g \in \mathcal{G}\) is unobservable, the true physical occurrence process corresponds to \(p = f(d, g)\). By implicitly marginalizing over the latent geology through dataset-level expectations, \(\pi(p \mid d) = \int \pi(p \mid d, g) \pi(g \mid d) \, dg\), the conditional distribution becomes multimodal. GeoCFM establishes a time-dependent transport velocity field in continuous 2D space from a standard Gaussian base distribution \(\pi_0 = \mathcal{N}(0, I_2)\) to empirical deposit locations, bridging dense multi-scale raster features and sparse spatial points via bilinear feature probing.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Multi-channel Geo-image d<br/>(Magnetics, Gravity, Geochemistry)"] --> B["UNet Dense Feature Extractor"]
    C["Base Gaussian Points z ~ N(0, I2)<br/>and Time Step t in [0, 1]"] --> D["Continuous Point Trajectory<br/>pt = t p + (1 - t) z"]
    B --> E["Bilinear Feature Sampling & Point Probing"]
    D --> E
    E --> F["Residual Velocity Head & ODE Numerical Solver"]
    F --> G["Stochastic Occurrence Points & Prospective Density<br/>(Epistemic Uncertainty via Repeated Draws)"]

Key Designs

1. Point-Wise Continuous Flow Matching: Eliminating Pseudo-Negative Bias via Marginal Transport Conventional prospectivity mapping forces binary classifiers or segmentation networks to treat unlabeled ground as negative labels, introducing severe decision boundary distortion. GeoCFM fundamentally redesigns the learning objective by treating each known deposit as a draw from the empirical data distribution \(p \sim \pi_{\text{data}}(\cdot \mid d)\) in continuous coordinate space \(p \in \mathbb{R}^2\). The model defines a time-dependent velocity field \(v_\theta(p, t \mid d)\) that pushes points along a straight-line interpolation trajectory between a base Gaussian sample \(z \sim \mathcal{N}(0, I_2)\) and a target deposit \(p\): \(p_t = t p + (1 - t) z\). Under flow matching, the analytical target velocity along this trajectory is constant: \(v^* = \dot{p}_t = p - z\). By directly regressing the predicted velocity against this target vector across all observed occurrences, GeoCFM learns the generative transport field entirely from positive locations without requiring any pseudo-negative sampling or heuristic background assumptions.

2. Image-to-Points Conditioning via Bilinear Feature Probing: Decoupling Coordinates from Raster Grids Conditioning a continuous 2D vector field on high-dimensional multi-channel raster inputs presents a significant structural challenge. GeoCFM introduces an image-to-points conditioning mechanism: a multi-layer UNet encoder \(F_\xi\) first extracts a dense multi-scale spatial feature tensor \(\Phi_\xi = F_\xi(d) \in \mathbb{R}^{B \times D \times H \times W}\), preserving regional magnetic gradients, lithological contacts, and structural lineaments. For any moving particle at coordinate \(p_t \in [0, 1]^2\) along the ODE trajectory, a continuous bilinear interpolation operator \(B(p_t)\) probes the local feature vector \(\phi = B(p_t)\Phi_\xi \in \mathbb{R}^D\) at the exact particle location. This continuous feature lookup decouples spatial precision from fixed raster pixel resolution and guarantees end-to-end differentiability across continuous space.

3. Sinusoidal Time Embedding and Residual Velocity Head: Stable ODE Numerical Integration Conditioned on the probed local geological context, GeoCFM concatenates the local feature vector \(\phi\), normalized particle coordinates \(p_t\), and a sinusoidal time embedding of \(t\). The concatenated representation is passed into a shared per-point residual network velocity head \(h_\psi(\phi, p_t, t)\), which outputs the 2D velocity vector \(\dot{p}_t = v_\theta(p_t, t \mid d) \in \mathbb{R}^2\). During inference on an unseen geo-image \(d\), the model samples \(N\) initial particles from \(\mathcal{N}(0, I_2)\) and numerically solves the probability flow ODE from \(t = 0\) to \(t = 1\) using an explicit Euler solver over \(K\) discrete integration steps: $\(p^{(k+1)} = p^{(k)} + \Delta t \, v_\theta\left(p^{(k)}, t^{(k)} \mid d\right), \quad t^{(k)} = k \Delta t, \quad k = 0, \ldots, K-1\)$ The resulting point sets localize tightly along favorable structural and geophysical contacts. Repeated sampling runs under the same observation uncover alternative prospective clusters, directly communicating epistemic non-identifiability to exploration geologists.

Loss & Training

GeoCFM is optimized end-to-end via a simulation-free conditional flow matching objective, minimizing the mean squared regression error against target displacement vectors: $\(\mathcal{L}(\theta) = \mathbb{E}_{d \sim \mathcal{D}} \, \mathbb{E}_{p \sim \pi_{\text{data}}(\cdot \mid d)} \, \mathbb{E}_{z \sim \mathcal{N}(0, I_2)} \, \mathbb{E}_{t \sim \text{Unif}[0, 1]} \left[ \left\| v_\theta(p_t, t \mid d) - (p - z) \right\|^2 \right]\)$ The network is optimized using AdamW with an initial learning rate of \(3 \times 10^{-4}\), weight decay of \(10^{-4}\), cosine annealing scheduling, and gradient clipping at norm 1.0. Inference employs \(K=50\) Euler integration steps by default, while ablation experiments confirm that as few as \(K=5\) steps provide highly competitive point-set fidelity.

Key Experimental Results

Main Results

The authors evaluated GeoCFM across two benchmark configurations: (1) a controlled Synthetic Benchmark representing porphyry-style mineralization with latent geochemical field activation; (2) real continental-scale USGS Earth Mapping Resources Initiative (Earth MRI) multi-channel geophysical grids paired with official MRDS/MAS-MILS mineral occurrence records, evaluated under a rigorous spatially disjoint tile split to eliminate spatial autocorrelation leakage. Performance is measured across 5 complementary metrics: Chamfer distance (CD, pixels), Sinkhorn optimal transport divergence (Sink.), 5-pixel tolerance F-score (F@5), Kernel Density Estimation negative log-likelihood (NLL), and top-5% most prospective pixel hit rate (Top5). All metrics reflect the mean and standard error across 20 stochastic draws per evaluation tile.

Table 1: Main experimental evaluation on Synthetic and USGS Earth MRI benchmarks (spatially disjoint test split)

Benchmark	Method	CD (px) ↓	Sinkhorn ↓	F@5 ↑	KDE-NLL ↓	Top-5% Hit ↑
Synthetic	Uniform Sampling	49.61 ± 2.28	0.147 ± 0.007	0.206 ± 0.015	10.804 ± 0.006	0.048 ± 0.006
	Global KDE	41.16 ± 2.04	0.121 ± 0.006	0.255 ± 0.014	10.583 ± 0.030	0.058 ± 0.008
	OCSVM (One-Class SVM)	45.56 ± 2.20	0.134 ± 0.007	0.246 ± 0.015	10.557 ± 0.020	0.236 ± 0.018
	Retrieval-KDE	47.16 ± 2.86	0.132 ± 0.007	0.220 ± 0.016	13.302 ± 0.440	0.058 ± 0.011
	Poisson-LR	39.93 ± 2.16	0.122 ± 0.006	0.384 ± 0.016	9.803 ± 0.042	0.500 ± 0.028
	Random Forest (RF)	37.09 ± 2.17	0.116 ± 0.006	0.462 ± 0.018	9.552 ± 0.054	0.586 ± 0.025
	GBDT	32.13 ± 2.03	0.103 ± 0.006	0.546 ± 0.018	9.288 ± 0.083	0.630 ± 0.027
	UNet-Seg	29.46 ± 2.18	0.100 ± 0.006	0.615 ± 0.022	9.143 ± 0.089	0.661 ± 0.031
	GeoCFM (Ours)	9.37 ± 0.62	0.045 ± 0.003	0.634 ± 0.012	8.950 ± 0.110	0.720 ± 0.020
Earth MRI	Uniform Sampling	64.25 ± 2.34	0.260 ± 0.008	0.052 ± 0.003	11.598 ± 0.288	0.047 ± 0.007
	Global KDE	66.35 ± 2.38	0.277 ± 0.008	0.050 ± 0.003	12.053 ± 0.303	0.043 ± 0.007
	OCSVM (One-Class SVM)	58.38 ± 2.33	0.240 ± 0.008	0.053 ± 0.003	11.946 ± 0.313	0.059 ± 0.008
	Retrieval-KDE	61.14 ± 2.34	0.250 ± 0.008	0.053 ± 0.003	11.698 ± 0.298	0.061 ± 0.009
	Poisson-LR	64.21 ± 2.26	0.255 ± 0.008	0.055 ± 0.004	11.274 ± 0.239	0.059 ± 0.008
	Random Forest (RF)	54.37 ± 2.23	0.225 ± 0.008	0.071 ± 0.004	11.504 ± 0.297	0.095 ± 0.011
	GBDT	52.98 ± 2.14	0.215 ± 0.007	0.069 ± 0.004	11.233 ± 0.269	0.102 ± 0.010
	UNet-Seg	46.81 ± 2.30	0.205 ± 0.008	0.085 ± 0.006	11.620 ± 0.290	0.130 ± 0.013
	GeoCFM (Ours)	12.30 ± 0.54	0.085 ± 0.004	0.151 ± 0.008	10.500 ± 0.150	0.250 ± 0.012

Ablation Study

The sensitivity of GeoCFM was rigorously analyzed across varying Euler numerical integration steps \(K\) and reduced proportions of available positive training deposits.

Table 2: Sensitivity to Euler ODE integration steps and positive label availability on Chamfer distance (CD, px)

Dimension	Experimental Configuration	Synthetic CD (px) ↓	Earth MRI CD (px) ↓	Analysis & Observation
Euler Steps \(K\)	\(K = 50\) (Default)	9.37 ± 0.62	12.30 ± 0.54	Full trajectory integration providing finest point mode concentration
	\(K = 20\)	9.34 ± 0.60	12.18 ± 0.52	Statistically identical performance to \(K=50\) within error bounds
	\(K = 10\)	9.32 ± 0.61	12.10 ± 0.53	80% reduction in ODE step compute with no metric degradation
	\(K = 5\)	9.30 ± 0.61	12.00 ± 0.55	Ultra-fast inference with optimal geometric retention
Available Positives	100% Positives	9.37 ± 0.62	12.30 ± 0.54	Baseline with all historical labeled deposits
	75% Positives (25% dropped)	9.48 ± 0.65	12.15 ± 0.58	Negligible variation under 25% missing exploration labels
	50% Positives (50% dropped)	9.75 ± 0.68	12.65 ± 0.62	Substantial margin of superiority maintained over all baselines

Key Findings

• Substantial Outperformance over Pseudo-Negative Baselines: On the challenging Earth MRI spatially disjoint split, GeoCFM achieves a Chamfer distance of 12.30 px compared to 46.81 px for the best discriminative baseline UNet-Seg—a relative error reduction of 73.7%. The top-5% hit rate nearly doubles from 13.0% to 25.0%, proving that generative transport learning without pseudo-negatives avoids the boundary distortion that plagues discriminative classifiers.
• Inference Efficiency and ODE Robustness: Computing UNet feature maps once per image tile amortizes computational costs; each subsequent Euler step requires only lightweight bilinear interpolation and small MLP forward passes. Decreasing Euler steps from \(K=50\) to \(K=5\) preserves a Chamfer distance of 12.00 px on Earth MRI, proving that straight-line transport trajectories allow fast Monte Carlo scenario sampling.
• Robustness to Severe Label Sparsity: With half of all known positive occurrences omitted (50% positive subset), test CD only shifts from 12.30 px to 12.65 px, establishing that the model captures genuine geophysical structural cues rather than memorizing individual deposit coordinates.

Highlights & Insights

• Conceptual Paradigm Shift: Successfully reframes mineral prospectivity from deterministic score-map regression into positive-only conditional generative sampling, reconciling the fundamental geoscientific premise that "absence of evidence is not evidence of absence".
• Elegant Hybrid Architecture: Harmoniously combines UNet dense multi-scale raster feature extraction with continuous point-wise flow matching via local bilinear probing, ensuring sub-pixel continuous precision decoupled from raster resolution.
• Explicit Epistemic Uncertainty Representation: Generates diverse, plausible mineral occurrence hypotheses from identical surface observations, exposing unobservable deep subsurface non-identifiability through spatial variance across independent draws.

Limitations & Future Work

• Preset Sampling Budget: The number of generated points \(N\) is currently a user-specified Monte Carlo budget rather than an adaptive, data-driven estimate of the true deposit count.
• Physics-Informed Structural Guidance: The velocity field is trained in a purely data-driven fashion without hard geometric constraints such as structural fault alignments or hydrothermal fluid-flow differential equations.
• Future Directions: Integrating 3D forward geological simulators with conditional flow matching and incorporating sparse drill-core logging constraints to achieve calibrated 3D regional mineral targeting.

Related Work & Insights

• vs. Discriminative Score-Map Models (GBDT, RF, UNet-Seg): Discriminative models rely on pseudo-negatives sampled from unlabeled regions, corrupting decision boundaries and producing single deterministic maps that fail under one-to-many conditions. GeoCFM learns a positive-only conditional point distribution, naturally supporting multi-modal target sampling.
• vs. One-Class Classifiers & Retrieval Baselines (OCSVM, Retrieval-KDE): One-Class SVM models only estimate positive support in feature space without spatial contextual reasoning (Earth MRI CD 58.38 px); Retrieval-KDE cannot generalize across complex multi-modal anomalies. GeoCFM achieves vastly superior geometric precision through deep UNet guidance.
• vs. Diffusion Models for Point Clouds: Conventional diffusion models require hundreds of discrete denoising steps and often suffer from trajectory curvature. Flow matching provides straight-line transport paths, yielding stable sampling in as few as 5 Euler steps.

Rating

• Novelty: ⭐⭐⭐⭐⭐ [Pioneering integration of positive-only conditional flow matching with spatial mineral prospectivity point sampling]
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ [Rigorous evaluation on synthetic latent-control benchmarks and USGS Earth MRI data under spatially disjoint tile splits]
• Writing Quality: ⭐⭐⭐⭐⭐ [Clear mathematical formulation transitioning smoothly from geological non-identifiability to point-flow mechanics]
• Value: ⭐⭐⭐⭐⭐ [Provides substantial methodological inspiration for geoscience, remote sensing, and positive-only spatial machine learning]

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