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

Moshe Eliasof, Eldad Haber

arXiv:2609.34398v1cs.LG

TL;DR

Mineral occurrences are sparse, unlabeled locations are not reliable negatives, and similar geo-images can correspond to different outcomes. GeoCFM samples occurrence point sets from a conditional spatial distribution, outperforming broad baselines across five metrics on synthetic and USGS Earth MRI data.

  • Problem

    Occurrences are sparse and unlabeled locations are not reliable negatives, while unobserved geology makes the mapping from geo-images to occurrences one-to-many.

    Source: §1 · p. 3, §3 · p. 6

  • Method

    GeoCFM learns a positive-only conditional spatial density and uses flow matching with point-conditioned UNet features to generate occurrence point sets from geo-images.

    Source: §1 · p. 3

  • Results

    GeoCFM captures structured one-to-many behavior and outperforms a broad baseline suite across all five evaluation metrics on synthetic and USGS Earth MRI data with a spatially disjoint split.

    Source: §6 · p. 15

  • Takeaways & Limitations

    Drawing multiple samples for the same geo-image exposes epistemic uncertainty induced by unobserved geology.

    Source: §6 · p. 15

    The latent geology is unobserved because deep-earth structures cannot be directly measured, so predictions concern the marginal distribution given geo-images.

    Source: §3.1 · p. 6

Abstract

from arXiv · show

Critical mineral discovery is a positive-only problem: deposits are observed as sparse locations, while unlabeled regions are not reliable negatives, and similar geophysical signatures can arise from different subsurface states. We therefore model mineral targeting as learning a conditional spatial distribution over occurrence locations, $\pi(p\mid d)$, given geo-images $d$, rather than predicting a deterministic per-pixel score map. We introduce GeoCFM, a conditional flow-matching model that generates mineral occurrence point sets conditioned on multi-channel geo-images; GeoCFM learns a point-wise transport field in $\mathbb{R}^2$, using UNet features with point-conditioned velocity prediction to bridge dense rasters and sparse supervision without pseudo-negatives. On a synthetic magnetics--geochemistry benchmark with latent activation and on USGS Earth MRI data with a spatially disjoint tile split, GeoCFM improves geometric agreement with observed occurrences over score-map and non-conditional baselines, while representing epistemic uncertainty through conditional sampling.

1 Introduction

Mineral exploration combines sparse known deposits with unlabeled regions that cannot be treated as reliable negatives, while similar measurements may reflect different subsurface states. GeoCFM addresses this by sampling conditional occurrence locations from geo-images, rather than producing a deterministic score map.

1 Introduction

  • The figure contrasts supervised learning with positive-and-negative labels, positive-only exploration data, and GeoCFM’s conditional-density approach.Its panels depict how sparse deposits and unlabeled areas motivate probabilistic mineral discovery maps from positive samples alone.Source: Figure 1 · p. 2
  • Known deposits are sparse positives, while unlabeled areas may include untested or undiscovered sites and are not trustworthy negatives.Sampling pseudo-negatives can bias decision boundaries, and historical exploration concentrates labels and supporting data in previously targeted areas.Source: §1 · p. 2, Figure 1 · p. 2
  • Similar geo-image measurements can correspond to multiple latent geological states, making mineral occurrence locations a one-to-many inference problem.The latent geology includes unobserved structure and geological history; this non-identifiability motivates probabilistic rather than deterministic predictions.Source: §1 · pp. 2–3
  • GeoCFM models a conditional spatial density over occurrence locations and generates point sets from geo-images using a point-conditioned velocity field with local UNet features.The approach uses positive occurrences without pseudo-negatives, bridging dense raster inputs and sparse occurrence supervision.Source: §1 · p. 3
  • GeoCFM is reported to capture one-to-many behavior and outperform broad baseline suites on synthetic and USGS Earth MRI data.The Earth MRI evaluation uses a spatially disjoint split; the contribution summary describes comparisons across score-map, boosting, one-class, retrieval, and non-conditional methods.Source: §1 · p. 4

2 Related Work

Prior prospectivity mapping commonly produces deterministic favorability scores and may use pseudo-negatives, while positive-only learning and conditional generative modeling offer alternatives. GeoCFM applies conditional flow matching to sample sparse occurrence locations from geo-images.

Mineral Discovery/Prospectivity mapping

  • Prospectivity mapping commonly produces a single deterministic score map, and pseudo-negatives can include undiscovered positives or amplify spatial bias.The paper contrasts those maps with a sampleable, multi-modal conditional spatial density and compares against score-map, boosting, one-class, and retrieval baselines.Source: §2 · p. 4

Positive-only and PU learning

  • Positive-unlabeled learning treats unlabeled data as a mixture, but its assumptions can conflict with exploration’s spatially biased surveying and drilling.GeoCFM instead models observed occurrences as samples from a conditional spatial density, avoiding explicit negative sampling.Source: §2 · pp. 4–5

Generative models and flow matching

  • Conditional flow matching learns a vector field for transport; GeoCFM uses geo-images to guide point-flow and generate sparse occurrence locations.This differs from image generation conditioned on labels and from geological work that generates dense 3D volumes.Source: §2 · p. 5

3 Mineral Discovery as a Learning Problem

Mineral discovery is framed as estimating occurrence locations from geo-images while accounting for unobserved geological context. A synthetic benchmark illustrates how hidden geochemical controls can make visually similar magnetic anomalies yield different occurrence patterns.

3 Mineral Discovery as a Learning Problem

  • The target is a conditional spatial density over occurrence coordinates, rather than a dense binary label map, because unlabeled locations are not reliable negatives.Occurrences are represented as points whose count can vary across geo-images.Source: §3 · p. 6

3.1 Implicit marginalization

  • The desired density marginalizes over latent geology, which is not directly observed or explicitly sampled; a parametric flow model instead learns to sample from π(p | d).The model is averaged over geo-images as an approximation to expectation over geological contexts.Source: §3.1 · p. 6

3.2 Synthetic magnetic-geochemistry model problem

  • In the synthetic benchmark, hidden geochemical conditions govern which magnetic rings activate and where occurrences localize, so similar anomalies can produce different, potentially multi-modal patterns.Figure 3 shows occurrences on only a subset of rings and along particular arcs.Source: §3.2 · p. 7, Figure 3 · p. 7

4 Mineral Discovery using Flow Matching

GeoCFM learns a conditional velocity field that transports points from a simple base distribution toward observed occurrence locations. It trains on interpolated base-to-data trajectories and conditions point-wise velocity predictions on local features from geo-images.

4 Mineral Discovery using Flow Matching

  • GeoCFM samples stochastic occurrence sets from a learned conditional density by integrating a velocity-field ODE from Gaussian points to the target distribution.The resulting set is conditioned on the geo-image rather than being a single deterministic prediction.Source: §4 · p. 7, §4.1 · p. 8

4.1 Conditional Flow Matching for Mineral Occurrence Locations

  • Conditional flow matching learns a time-dependent velocity field that transports a simple base distribution to the observed occurrence distribution.The model uses a two-dimensional standard Gaussian as its base distribution.Source: §4.1 · p. 8

4.2 Training Objective

  • Training linearly interpolates a Gaussian sample and an observed occurrence, then regresses predicted velocity to the constant target p - z.The objective minimizes mean-squared error across data points, base samples, and interpolation times.Source: §4.2 · p. 8

4.3 Conditioning on Image-Like Observations

  • A UNet encodes each geo-image, bilinear sampling extracts local features at each noised point, and a shared per-point network predicts its two-dimensional velocity.The local feature, point coordinate, and time embedding inform the velocity prediction.Source: §4.3 · pp. 8–9, Figure 4 · p. 9

5 Experiments

Across synthetic and Earth MRI benchmarks, GeoCFM outperforms the evaluated baselines in conditional occurrence sampling. Experiments also examine one-to-many samples, generalization to held-out geographic tiles, and sensitivity to available positives and sampling cost.

5 Experiments

  • The evaluation maps each baseline’s score or density map into a spatial distribution and draws the same number of points per patch as GeoCFM.It uses five metrics and averages each over 20 independent draws per input.Source: §5 · p. 10

5.1 Synthetic Magnetics–Geochemistry Benchmark

  • Four GeoCFM draws for one synthetic input concentrate along contact zones while activating only a subset of visually similar rings.The benchmark’s latent activation mechanism makes occurrence locations depend on hidden controls.Source: §5.1 · pp. 11–12, Figure 6 · p. 12

5.2 USGS Earth MRI Benchmark

  • GeoCFM achieves the best value on all five synthetic metrics and all five Earth MRI test metrics, with real-data gains generalizing to new geographic tiles.The synthetic evaluation averages 20 draws over 50 held-out examples; Earth MRI averages 20 draws across 200 test patches.Source: §5.1 · p. 12, §5.2 · pp. 13–14, Table 1 · p. 13

5.3 Ablations and Analyses

  • Reducing training positives from 100% to 50% keeps synthetic CD within 9.30–9.75 px and Earth MRI CD within 11.76–12.65 px across three seeds.The analysis distinguishes sensitivity to label availability from predictive sampling uncertainty.Source: §5.3 · p. 14
  • With K = 5 Euler steps, CD is 9.30 on synthetic data and 12.00 on Earth MRI, within the standard deviation of K = 50.The UNet feature map is computed once per patch; each Euler step uses a bilinear lookup and lightweight velocity head.Source: §5.3 · p. 14
  • Larger point-sampling budgets N improve KDE-NLL and density stability on both benchmarks, with diminishing returns past a few hundred samples.N is a Monte Carlo budget for the conditional density, not an estimate of the true deposit count.Source: §5.3 · p. 14

6 Summary and Outlook

GeoCFM reframes prospectivity mapping as conditional occurrence-location sampling from positive-only supervision, and improves agreement with observed occurrences across five metrics on synthetic and Earth MRI data. Sampling also makes ambiguity explicit and supports uncertainty-aware target ranking.

Summary

  • Across synthetic magnetics–geochemistry and spatially disjoint USGS Earth MRI evaluations, GeoCFM captures structured one-to-many behavior and outperforms a broad baseline suite on all five metrics.The paper proposes future extensions with geological priors, adaptive occurrence-count estimation, larger multimodal datasets, and forward simulators.Source: §6 · p. 15
  • GeoCFM learns a conditional spatial distribution over occurrence locations from positive-only supervision, using point-conditioned UNet features without pseudo-negatives.Multiple samples for the same geo-image expose epistemic uncertainty associated with unobserved geology.Source: §6 · p. 15

Outlook

  • Probabilistic sampling makes ambiguity explicit and enables uncertainty-aware target ranking, while future extensions could add stronger geoscientific priors and regional-scale scenario generation.The outlook names constrained masks, structural-contact penalties, physics-informed objectives, geological forward simulators, and adaptive occurrence-count estimation.Source: §6 · p. 15

A Additional Experimental Details

The appendix adds implementation, benchmark, preprocessing, baseline, and evaluation details, along with more held-out Earth MRI comparisons.

Additional Experimental Details

B Implementation Details for GeoCFM

GeoCFM learns a conditional velocity field that transports Gaussian samples toward occurrence locations, conditioning point-wise predictions on UNet image features. Training and sampling use normalized coordinates and explicit numerical procedures.

B Implementation Details for GeoCFM

  • GeoCFM learns a conditional velocity field transporting base-Gaussian samples to the occurrence-location distribution, trained by matching the velocity along a linear interpolation path.The target velocity is x_1-z, and the objective is its squared distance from the predicted velocity.Source: B Implementation Details for GeoCFM · p. 19

B.1 Image-conditioned point transport

  • A UNet extracts dense image features; bilinear sampling at each noised point combines local features, point coordinates, and a time embedding for velocity prediction.A shared point-wise network predicts velocity, combining multi-scale image context with information at the current location.Source: §B.1 · p. 20

B.2 UNet backbone and velocity head

  • The image encoder uses residual convolutional blocks, while the velocity head is an MLP on Earth MRI and a residual fully connected head in the final synthetic experiments.The same UNet template is used across benchmarks with benchmark-specific widths.Source: §B.2 · p. 20

B.3 Coordinate handling

  • Coordinates are normalized to [0, 1]^2 for training and ODE integration, then mapped back to pixels using patch dimensions.Source: §B.3 · p. 20

B.4 Input normalization

  • Each input channel is standardized independently using non-zero pixels, leaving zero-filled no-data regions unchanged.The same normalization convention is used for both synthetic and Earth MRI experiments.Source: §B.4 · p. 20

B.5 Optimization and sampling

  • At test time, explicit Euler integration advances samples in K steps, and final samples are clipped to the valid image domain.Training uses AdamW, cosine learning-rate decay, gradient clipping, and GPU acceleration.Source: §B.5 · p. 21

C Synthetic Magnetics-Geochemistry Benchmark

The benchmark and appendix detail synthetic and Earth MRI data construction, a spatially disjoint evaluation split, point-sampling baselines, and geometric evaluation. Additional held-out Earth MRI patches compare GeoCFM samples with observed occurrences and baseline outputs.

C Synthetic Magnetics-Geochemistry Benchmark

  • The synthetic benchmark conditions on observed magnetics and geochemistry while withholding a latent geochemical field that activates only some candidate bodies, creating one-to-many occurrence patterns.Deposits within active bodies localize along spatially concentrated arcs.Source: C Synthetic Magnetics-Geochemistry… · pp. 21–22

C.1 Training and validation protocol

  • Synthetic examples are generated online during training, with validation on 16 fixed held-out examples; the reported experiments use the final UNet and residual velocity-head model.Source: §C.1 · p. 22

D.1 Raster construction; D.2 Gap filling and no-data handling; D.3 Occurrence alignment; D.4 Patch extraction

  • Earth MRI geophysical layers are rasterized to a common grid, gaps are filled by iterative inverse-distance weighting, and unsupported pixels remain zero as no-data regions.MRDS occurrences are aligned to the raster; retained patches require at least 50% valid-footprint pixels and use 128 × 128 windows.Source: §D.1 · p. 22, §D.2 · p. 22, §D.3 · p. 23

D.5 Spatially disjoint train-test split

  • Earth MRI train and test patches are assigned by 384 × 384 spatial tiles, with 20% held out and boundary-straddling patches discarded under a buffer rule.This is stricter than assigning patches by their top-left corners alone.Source: §D.5 · p. 23

E Baseline Details

  • Baselines use a shared point-sampling framework; they include unconditional global KDE, pixel-wise classifiers trained with pseudo-negatives, a UNet score-map model, and positive-only OCSVM.The UNet segmentation baseline shares GeoCFM's backbone family but replaces point-conditioned transport with a single-channel segmentation head.Source: E Baseline Details · p. 23, §E.2 · p. 24, §E.3 · p. 25

D.6 Additional qualitative comparisons on Earth MRI

  • On held-out Earth MRI patches, GeoCFM samples are more concentrated around observed structures than unconditional or score-map baselines while retaining uncertainty across repeated draws.Figure 1 rows are patches; columns show ground truth, GeoCFM, and the listed baselines, with observed occurrences and samples marked separately.Source: §D.6 · p. 23, Figure 1 · p. 24

E.6 GBDT; E.8 Retrieval-KDE

  • Retrieval-KDE estimates a density from retrieved training occurrences without a learned geo-image encoder, while GBDT converts pixel-wise scores into sampled point locations.The baselines generate as many points as observed deposits for each queried image or patch.Source: E Baseline Details · p. 23, §E.8 · p. 26, §E.6 · p. 26

F.1 Chamfer distance

  • Methods are evaluated with symmetric Chamfer distance between generated and observed point sets, computed in pixel coordinates.Pairwise distances are accumulated in float64 for numerical robustness.Source: §F.1 · p. 26

F.2 Additional metrics

The section defines complementary measures and sampling protocols for evaluating generated occurrence sets, then reports train–test Chamfer distances as evidence of generalization on both benchmarks.

F.2 Additional metrics

  • Sinkhorn divergence, F@5, KDE negative log-likelihood, and top-5% hit rate complement Chamfer distance by assessing distribution matching, point matching, density likelihood, and prospective-area coverage.Metrics are averaged over 20 stochastic draws per input and reported as mean ± standard deviation across held-out examples.Source: §F.2 · p. 27

F.3 Point-count protocol

  • Each method generates exactly as many points as the observed deposits in an image or patch, focusing comparisons on spatial quality rather than point-count mismatch.Source: §F.3 · p. 27

F.4 Repeated draws

  • Qualitative results use repeated independent draws for each input, with 500 sampled points per synthetic draw and 300 per Earth MRI draw.Source: §F.4 · p. 27

F.5 Train–test generalization gap

  • GeoCFM’s train–test Chamfer distances are 8.81 →9.37 on synthetic data and 11.89→12.30 on Earth MRI, supporting genuine conditional generalization.The reported gaps are comparable in relative terms to baselines despite GeoCFM’s much lower absolute error.Source: §F.5 · p. 27

G Additional Ablation Results

The appendix provides full tables for positive-subset stability, inference cost across Euler-step counts, and sampling-budget effects on density quality and stability.

Positive-subset stability

Inference cost

Sampling budget

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