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ML-Driven Strong Lens Discoveries: Down to $\theta_E \sim 0.03''$ and $M_\mathrm{halo}< 10^{11} M_\odot$

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that machine-learned classifiers trained on realistic simulations can find gravitational lenses with Einstein radii down to about 0.03 arcseconds and halo masses below 10^11 solar masses, a regime that is effectively…

desk verdict A careful simulation-driven forecast of low-mass strong lenses with JWST, but the headline 100% purity is not demonstrated and one table contradicts another. read the letter →

arxiv 2507.01943 v1 pith:PO3ZVGI7 submitted 2025-07-02 astro-ph.CO astro-ph.GA

Ethan Silver , R. Wang , Xiaosheng Huang , A. Bolton , C. Storfer , S. Banka This is my paper
classification astro-ph.COastro-ph.GA
keywords stronggravitationallensingJWSTmachinelearningResNetU-Netlow-massdarkmatterhalosEinsteinradiusCDMtests
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that machine learning, trained on simulations, can extend strong gravitational lens searches into a new regime: lenses with Einstein radii as small as 0.03 arcseconds, produced by halos below $10^{11}M_\odot$, which human inspectors cannot see. It forecasts that JWST can find about 17 such low-halo-mass lenses per square degree, and that a combined ResNet-plus-U-Net pipeline can localize about 1.1 per square degree at 100 percent precision, or about 7 per square degree at 99 percent precision. For conventional lenses ($\theta_E>0.5''$), the same ResNet reaches near-100 percent completeness and purity on simulated and real HST images, and it found two HST lens candidates that a crowdsourced human search missed. The payoff is a new way to test cold dark matter, because halo abundance below the typical galactic scale is where CDM predictions are largely untested.

What carries the argument

The machinery is a two-stage pipeline. First, a 'shielded' ResNet classifier scans image cutouts and assigns each a lens probability; it was trained on thousands of simulated images built from CosmoDC2 lens halos, VELA hydrodynamic source galaxies, and Sersic/environmental galaxies, with HST or JWST noise added during training. Second, a U-Net segmentation model takes ResNet-positive cutouts and predicts a per-pixel lens-location probability, so the lens position can be pinpointed. The forecasts that set the discovery rates use the CosmoDC2 halo catalog plus the Zahid et al. (2018) halo-mass-to-velocity-dispersion relation to convert halo masses into Einstein radii, and count lens-source overlap with three geometric methods: source in lens area, lens in source area, or either.

What would settle it

Run the trained RUN pipeline on a few square degrees of deep JWST imaging; if the yield of localized lenses with halo mass below $10^{11}M_\odot$ and Einstein radii between $0.02''$ and $0.05''$ is far below 1.1 per square degree at the claimed precision threshold, the forecast and transferability assumption fail.

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Extended reading notes

Core claim

The central claim is that strong lens detection is no longer limited by human visual inspection or by Einstein radii large enough to show visible arcs: a shielded ResNet classifier trained on simulations with realistic hydrodynamic sources can identify lenses down to JWST's diffraction limit, and a U-Net can pinpoint their locations. On simulated JWST data with 10,000-second exposures, the ResNet achieves high classification accuracy for lenses with $0.02''<\theta_E<0.15''$, and the combined RUN pipeline reaches 100 percent pixel-level precision at a 0.96 ResNet threshold and a 0.72 U-Net threshold. Applied to real HST data, the conventional-lens model classified every test lens as a lens with high probability and identified two candidates missed by a crowdsourced citizen-science search. The authors conclude that the bottleneck is no longer finding candidate lenses but confirming them spectroscopically.

Load-bearing premise

The forecast numbers rest on the assumption that the simulated images, built from VELA sources, CosmoDC2 lenses, SIE mass profiles, and the JWST noise model, capture the appearance of real small-Einstein-radius lenses, and that the unmeasured halo-mass-to-velocity-dispersion relation below about 100 km/s used to convert halo mass into Einstein radius is roughly right.

Editorial extensions

If this is right

  • Near-100 percent completeness and purity for conventional lenses means space-based surveys with HST, JWST, Roman, and Euclid can automate lens discovery instead of relying on human scanning.
  • JWST should reveal about 17 lenses per square degree with halo mass below $10^{11}M_\odot$ and Einstein radii $0.02''$ to $0.05''$, with about 1.1 per square degree localized by the RUN pipeline at zero false positives.
  • The two HST candidates show that even comprehensive crowdsourced searches miss discoverable lenses; machine learning can recover such systems.
  • The low-mass lens population provides a new observational handle on CDM predictions for halo abundance below about $10^{11}M_\odot$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If real JWST images match the simulation inputs, then existing deep JWST surveys already contain these small lenses, and rerunning the trained pipeline on archival data is a direct test of the forecast.
  • Because the halo-mass-to-velocity-dispersion relation is unconstrained below about 100 km/s, the forecast rates could shift by an order of magnitude; measuring that relation from dwarf-galaxy kinematics would firm up or revise the numbers.
  • The same approach may extend to even smaller, effectively dark halos below $10^{10}M_\odot$ with higher-resolution instruments, since the lensing signal is not obscured by lens light.
  • Multi-band imaging, by separating lens and source light, could help confirm small-Einstein-radius candidates and improve precision beyond single-band detection.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper forecasts the number of strong gravitational lenses detectable with JWST, simulates lensed and unlensed images using CosmoDC2, JAGUAR/W18, and VELA sources, trains a shielded ResNet for three Einstein-radius ranges, and trains a U-Net to localize small-Einstein-radius lenses. The central new claims are that JWST can find ~17 deg^-2 lenses with 0.02''<θ_E<0.05'' and M_halo<10^11 M_sun using a ResNet, and that the combined ResNet+U-Net pipeline can localize ~1.1 deg^-2 of them at ~100% pixel-level precision (or ~7.0 deg^-2 at 99% precision). The paper also reports two new HST lens candidates missed by the Garvin et al. (2022) crowdsourcing search.

Significance. If the small-lens forecasts hold, this work opens a genuinely new observational regime: galaxy-scale strong lensing below θ_E≈0.03'' and M_halo≈10^11 M_sun would provide a new probe of low-mass dark matter halos. The simulation effort is unusually thorough for this type of forecast, and the k-fold VELA split, truncated-SIS test, environmental-galaxy variation, and real-HST validation for conventional lenses are genuine strengths. The two new HST candidates and the demonstration that Model 1a separates real lenses from non-lenses are concrete, falsifiable results that support the conventional-lens part of the paper. The small-lens part is scientifically important but currently rests on in-sample validation and unpropagated systematics.

major comments (4)
  1. [§3.3.3, Table 9 vs. Table 8] The forecast base for the headline bin is internally inconsistent. Table 8 states that the 0.02''<θ_E<0.05'' bin contains 240 lenses/deg^2 with M_halo<10^11 M_sun (the difference of the first two rows of Table 2), while Table 9 lists 490 lenses/deg^2 for the same bin and the same mass cut. Because the 17 and 1.1 deg^-2 numbers are derived from this bin, the authors must determine which base is correct, correct the affected tables/figures, and recompute the predicted yields.
  2. [§3.2.2.4, §4.2.2, Table 9, Abstract] The claim of ~100% precision for the 1.1/deg^2 (and 7.0/deg^2 at 99%) forecast is a validation-set zero-count statement, not a statistically supported field-deployment bound. With ~2000 Type-2 non-lenses, zero false positives gives a 95% upper limit of order 3.7 false positives (FPR≈0.0019); combined with the paper's own 1/3800 lens prior and a recall of ~0.67, the expected precision at the chosen ResNet threshold is not 100%. The paper itself computes a 5.56:1 FP:TP ratio at threshold 0.95 in §3.2.2.4, and the 0.96 and 0.72 thresholds were selected after inspecting the same validation set. The abstract and Table 9 should report a confidence interval for precision or an expected field precision under the stated prior, not a point value of 100% from zero counts.
  3. [§2.1.3, §2.3, §3.3.3] The headline yields (17, 7.0, and 1.1 deg^-2) are point estimates with no propagated uncertainty. The M_halo–σ relation of Zahid et al. (2018) is unconstrained below σ≈100 km/s, as the paper acknowledges in §2.3, and θ_E scales as σ^2; plausible changes in the normalization, slope, scatter, or low-σ behavior can shift the small-θ_E counts by an order of magnitude. The authors should provide credible intervals by varying the Z18 relation parameters, its scatter, and the VDF assumptions, and propagate these through Figure 21 and the U-Net recall factors.
  4. [§3.1, §3.2.2, §4.2, §3.3.3] The small-lens completeness and purity used in the forecast are measured on simulations generated with the same source catalog (VELA), lens profile (SIE), PSF model, and noise model used to construct the forecast, and no real JWST small-θ_E validation exists. The k-fold VELA split and truncated-SIS tests are useful robustness checks, but they do not constrain performance on real JWST images with realistic PSF structure, blending, and source morphologies at θ_E≈0.03''. The deployment rates should be explicitly framed as simulation-based predictions, and a concrete validation path—such as a blinded injection test on real JWST imaging or a pilot search with spectroscopic follow-up—should be identified.
minor comments (5)
  1. [§3.1.5] The exposure times for Model 2a and Model 2b appear swapped in the text: Table 3 assigns 10,000 s to Model 2a (JWST-long) and 1,000 s to Model 2b (JWST-short), but §3.1.5 states texp=1000 s for Model 2a and texp=10,000 s for Model 2b.
  2. [§3.2.1.1, Table 6] Table 6 lists the recall at probability threshold 0.5 as 0.9034, which is an outlier relative to the neighboring thresholds (0.9824 and 0.9794) and inconsistent with the text's statement of 0.980 recall at a 0.5 threshold; this is likely a typographical error and should be corrected.
  3. [§3.2.2.4] The deployment calculation contains a wording error: after estimating 1.8 true positives and 10 false positives, the text says '1.8 are false positives,' but the context indicates it should read '1.8 are true positives.'
  4. [§4.2.2 and §4.3] The text claims zero false positives for the U-Net above threshold 0.72 in one place, while §4.2.2 reports one false positive at that threshold and Figure 25 shows a red point above the 0.72 line; the 'zero' claim should be restricted to the ResNet-filtered subset used for the final pipeline, or the count should be stated consistently.
  5. [Abstract and §3.2.2.2] The abstract describes going down to θ_E≲0.03'', but Figure 14 shows that recall degrades sharply toward θ_E=0.02'' (roughly 40% at threshold 0.5); the completeness caveat at the lowest Einstein radii should appear near the headline claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lens forecasts are a forward model and the ML performance numbers are measured, not fitted parameters renamed as predictions.

full rationale

The claimed derivation chain is a forward model. Section 2 computes lens-source overlap counts from CosmoDC2 halos, the Z18 mass-velocity dispersion relation, SIE Einstein radii, and JAGUAR source densities; Sections 3-4 train ResNet/U-Net on simulated images and measure recall/precision on a held-out validation set; Tables 8-9 then multiply the forecast counts by the measured recalls in the relevant theta_E and M_halo bins. No equation fits a parameter to the quantity being predicted, and the ML metrics are applied after, not used as inputs to, the geometric lens count. The H21 and Storfer et al. citations provide the architecture and prior search context, but the networks are retrained and evaluated here, including out-of-sample checks (k-fold VELA splits, truncated SIS, real HST images), so they are not load-bearing self-citations. The choice of a 0.96 ResNet threshold that gives zero false positives on the validation set, and a 0.72 U-Net threshold with one false positive, is an in-sample purity statement with limited Poisson power; the paper itself notes that 'given the size of the validation set, Poisson noise could still permit false positives.' That is a statistical/external-validity caveat about the deployment purity claim, not a circular derivation. The discrepancy between the 240 and 490 lens/deg^2 forecast entries for 0.02''<theta_E<0.05'' is an internal-consistency/correctness issue, not a circularity. Overall the prediction does not reduce to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The forecast and ML results depend on several domain assumptions and hand-chosen thresholds. The most load-bearing are the Z18 halo mass to velocity dispersion relation extrapolated to low masses, the simulation-to-real transfer, and the probability thresholds chosen to yield zero false positives in validation.

free parameters (5)
  • Z18 Mhalo-sigma relation parameters (alpha, beta) = alpha=0.16, beta=3.31, scatter 0.17 dex
    Adopted to convert CosmoDC2 halo masses to velocity dispersion (Equation 2). The entire forecast scales with this relation, which is extrapolated below sigma ~ 100 km/s with no observational constraint.
  • ResNet classification threshold = 0.96
    Chosen to reach 100 percent precision on the validation set with Type 2 non-lenses (Table 7). Directly sets the 17 per square degree detectability forecast.
  • U-Net detection threshold = 0.72
    Chosen to achieve zero false positive pixel detections on the validation set (Figure 26). Sets the 1.1 per square degree pipeline forecast.
  • Elliptical selection B/T cut = 0.9
    Hand-chosen threshold to select lens galaxies from CosmoDC2 (Section 2.1.3). Affects the lens population and forecast counts.
  • Source detection threshold = 5 sigma_BKG
    Pixels above 5 sigma define observable source area (Method B). Controls counting of small-theta_E lens systems.
assumptions (6)
  • domain assumption SIE mass profile with sigma from Z18 adequately models low-mass lens halos.
    Used in Sections 2.1.3 and 3.1.1 to compute theta_E from halo mass. Real low-mass halos may have different density profiles, which would change detectability.
  • domain assumption VELA simulations are representative of high-redshift source galaxies.
    Sources are drawn from 34 VELA galaxies (Section 3.1.1). K-fold tests in Appendix D show limited overfitting, but real morphology diversity is larger.
  • domain assumption JWST PSF can be approximated as a Gaussian truncated at 3 sigma.
    Stated in Section 3.1.1. Real JWST PSF has wavelength-dependent structure, which could affect small-theta_E detection rates.
  • domain assumption Gaussian plus Poisson noise with W18 backgrounds reproduces HST and JWST observations.
    Used in Section 3.1.5. The HST validation for Model 1 supports this for large lenses, but the small-lens regime is untested.
  • domain assumption The source catalog JAGUAR and W18 is complete down to 29th magnitude.
    Underpins the source density in the forecast (Section 2.1.1). Incompleteness or Sersic-only morphologies would bias counts.
  • domain assumption The Z18 halo mass to velocity dispersion relation is valid down to 1e10 Msun.
    Needed for the Mhalo < 1e11 subset of the forecast. The paper acknowledges VDF uncertainty but does not quantify its impact on the headline numbers.

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Cite this review

Pith. "Pith review of ML-Driven Strong Lens Discoveries: Down to $\theta_E \sim 0.03''$ and $M_\mathrm{halo}< 10^{11} M_\odot$." pith.science (2026). https://pith.science/paper/PO3ZVGI7

@misc{pith2026250701943,
  author       = {Pith},
  title        = {Pith review of: ML-Driven Strong Lens Discoveries: Down to $\theta_E \sim 0.03''$ and $M_\mathrmhalo< 10^11 M_\odot$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO3ZVGI7}},
  note         = {Machine review of arXiv:2507.01943}
}
abstract

We present results on extending the strong lens discovery space down to much smaller Einstein radii ($\theta_E\lesssim0.03''$) and much lower halo mass ($M_\mathrm{halo}<10^{11}M_\odot$) through the combination of JWST observations and machine learning (ML) techniques. First, we forecast detectable strong lenses with JWST using CosmoDC2 as the lens catalog, and a source catalog down to 29th magnitude. By further incorporating the VELA hydrodynamical simulations of high-redshift galaxies, we simulate strong lenses. We train a ResNet on these images, achieving near-100\% completeness and purity for ``conventional" strong lenses ($\theta_E\gtrsim 0.5''$), applicable to JWST, HST, the Roman Space Telescope and Euclid VIS. For the first time, we also search for very low halo mass strong lenses ($M_{halo}<10^{11}M_\odot$) in simulations, with $\theta_E\ll 0.5''$, down to the best resolution ($0.03''$) and depth (10,000~sec) limits of JWST using ResNet. A U-Net model is employed to pinpoint these small lenses in images, which are otherwise virtually impossible for human detection. Our results indicate that JWST can find $\sim 17$/deg$^2$ such low-halo-mass lenses, with the locations of $\sim 1.1$/deg$^2$ of these detectable by the U-Net at $\sim100$\% precision (and $\sim 7.0$/deg$^2$ at a 99.0\% precision). To validate our model for finding ``conventional" strong lenses, we apply it to HST images, discovering two new strong lens candidates previously missed by human classifiers in a crowdsourcing project (Garvin et al. 2022). This study demonstrates the (potentially ``superhuman") advantages of ML combined with current and future space telescopes for detecting conventional, and especially, low-halo-mass strong lenses, which are critical for testing CDM models.

Figures

Figures reproduced from arXiv: 2507.01943 by the authors.

Figure 1
Figure 1. Random cutouts from the Williams et al. (2018) catalog around sources with flux > 5σ extract the pixels for each source.3 If the source has at least one pixel with I > 5σBKG, then that source is counted as a potentially observable source. This counting gives the total density of sources on the sky in each redshift bin. 2.1.2. Counting methods For the lenses, the total fraction of the area within the Einstein radius … view at source ↗
Figure 2
Figure 2. Examples of different types of systems for the purposes of counting (here we show simulated lensing systems from those generated in Section 3.1). The letters in the upper left corners correspond to the criteria under which the system would be counted to contribute to the “lens-able area”, explained in § 2.1.2. Image 3 shows cross-hairs pointing at the lens. Note that image 1 is not a double source: it has one source… view at source ↗
Figure 3
Figure 3. Distribution of σv given by Collett (2015) and CosmoDC2 using Zahid et al. (2018) on 1 deg2 of sky (CosmoDC2 provides halo mass, which is converted to stellar velocity dispersion σv using Equation 2). in CosmoDC2 catalog and used Z18 for conversion from halo mass to velocity dispersion. But they only selected halos with a stellar mass M∗ > 1011M⊙. Also, instead of SIE, they used the sum of the dark matter (modeled a… view at source ↗
Figures from the paper (43 more)
Figure 4
Figure 4. Figure 4: Distributions of zl , zs, and θE for the three Einstein radius ranges. for strong lensing systems to have lenses and sources that fall outside these ranges, especially with JWST. A simple and useful extension of this work would be to apply the same techniques to broade…
Figure 5
Figure 5. Figure 5: Flowchart describing the steps and sources used to simulate lensed and unlensed images in this training sample. To simulate unlensed images: for 0.5 ′′ < θE < 1.5 ′′, we turn off source light and set θE = 0; for θE < 0.50′′, we still set θE = 0, but keep the source lig…
Figure 6
Figure 6. Figure 6: Examples of simulated lensed systems with 0.5 ′′ < θE < 1.5 ′′ (Model 1 (HST-long/short)). Noise is added as described in § 3.1.5 at the level of Model 1a (HST-long) ( [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Validation and training loss and area under the ROC curve (AUC) over 360 epochs for Model 1a (HST-long). The validation AUC for the validation set reaches a maximum of 0.9978. We also indicate the best validation AUC and loss. The curves are boxcar smoothed with a wind…
Figure 8
Figure 8. Figure 8: Precision and recall for Model 1a (HST-long) as a function of probability threshold, with selected values shown in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Trained Model 1a (HST-long) tested on lenses with brighter arcs (top two rows) and non￾lenses (bottom two rows). From HST GO-15923, with exposure times between 4800 and 7800s. The model predicted probability is shown for each image. To further evaluate the validation p…
Figure 10
Figure 10. Figure 10: Trained Model 1a (HST-long) tested on lenses with fainter arcs (top row) and non-lenses (bottom row). From HST-GO 10886, with exposure time between 1500 and 2000s. The probability the model predicted is shown for each image. The results of these two tests therefore in…
Figure 11
Figure 11. Figure 11: AUC and loss curves for the model trained on small Einstein radius systems (0.15′′ < θE < 0.50′′) with texp = 10, 000 sec (Model 2a (JWST-long)) [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: AUC and loss curves for the model trained on small Einstein radius systems (0.15′′ < θE < 0.50′′) with texp = 1000 sec (Model 2b (JWST-short)). 3.2.2.2. Model 3 (JWST-small): validation performance Given that the performances for simulated systems with much smaller Ei…
Figure 13
Figure 13. Figure 13: AUC and loss curves for Model 3 (JWST-small) with 0.02′′ < θE < 0.15′′. For the non-lenses, light for the foreground galaxy is included in the simulation. This makes the training more challenging (see § 3.1.2) [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: Performance of Model 3 (JWST-small) with 0.02′′ < θE < 0.15′′ as a function of Einstein radius, at a threshold of 0.5. The columns at θE = 0.0 ′′ correspond to non-lenses. Therefore, on the left plot, the column at θE = 0.0 ′′ shows the fraction of non-lenses misclass…
Figure 15
Figure 15. Figure 15: Correctly classified lenses for Model 3 (JWST-small) (0.02′′ < θE < 0.15′′), or true positives. Four examples are shown, with one system in each row. The four columns show A. the full image seen by the ResNet, B. the image with the lens light removed, C. the image wit…
Figure 16
Figure 16. Figure 16: Incorrectly classified lenses for Model 3 (JWST-small) (0.02′′ < θE < 0.15′′), or lenses misclas￾sified as non-lenses (false negatives). The arrangement of the columns and the information shown on the left are the same as for [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: Incorrectly classified non-lenses for Model 3 (JWST-small) (0.02′′ < θE < 0.15′′), or non-lenses misclassified as lenses (false positives). The arrangement of the columns and the information shown on the left are nearly the same as for [PITH_FULL_IMAGE:figures/full_f…
Figure 18
Figure 18. Figure 18: Precision and recall for Model 3 (JWST-small) as a function of probability threshold, with selected values shown in [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: The forecast mass distribution of halos with 0.02′′ < θE < 0.15′′. Note that these histograms make it clear that the condition of Mhalo < 1011M⊙ essentially implies 0.02′′ < θE < 0.05′′. Though the inverse is not true: there are many systems with θE in that smallest b…
Figure 20
Figure 20. Figure 20: Two examples of simulated systems (in the context of Model 3 (JWST-small)) with a similar Einstein radius around 0.03′′. The left system has a low mass, and the right system has a mass almost two orders of magnitude larger. This is because the left system has a redshi…
Figure 21
Figure 21. Figure 21: These four panels illustrate the numbers we forecast for low mass lenses (Mhalo < 1011M⊙) that are detectable. The top row shows histograms of the overall forecast of lens numbers (using the methods from Section 2), as a function of Einstein radius. The top left panel…
Figure 22
Figure 22. Figure 22: The left half of this figure shows four examples of simulated systems where a 1% to 95% pixel scaling allows for clearly identifying the lens. The right half of this image shows two examples of simulated systems where the lens cannot be clearly identified with a 1% to…
Figure 23
Figure 23. Figure 23: Two new lens candidates found in this work. The first is at α = 09:36:03.00, δ = 09:14:03.12 (HST-144.0125+9.2342) and the second is at α = 12:02:00.41, δ = 47:42:16.92 (HST-180.5017+47.7047). Note that our naming convention uses the decimal coordinates of the lens ce…
Figure 24
Figure 24. Figure 24: Loss and accuracy vs. training epochs. The loss curve drastically drops in the first 25 epochs. It appears it may still be gradually decreasing afterward, as the accuracy continues to slowly improve. In our experiments, beyond approximately the 100th Epoch, the valida…
Figure 25
Figure 25. Figure 25: Detection Probability vs. Einstein radius at U-Net training epoch 110, for the Einstein radius range of 0.02′′ < θE < 0.15′′ (as shown on the x-axis) and all halo mass values. For an image with a lens, the pixel with the maximum probability represents the U-Net’s pred…
Figure 26
Figure 26. Figure 26: Pixel Level Precision vs. Recall curve for various ResNet thresholds chosen at Step 2 of the ResNet and U-Net pipeline, for the Einstein radius range of 0.02′′ < θE < 0.15′′ and all halo mass values. Colored shapes represent U-Net probability thresholds that are multi…
Figure 27
Figure 27. Figure 27: We show the forecast numbers for lenses with M < 1011M⊙ that are correctly classified by the ResNet Model 3 (JWST-small) at a 0.96 threshold (with zero false positives; see bottom right panel in [PITH_FULL_IMAGE:figures/full_fig_p044_27.png]
Figure 28
Figure 28. Figure 28: The blue histogram shows small Einstein radius lenses, with Mhalo < 1011M⊙ and θE < 0.05′′ , correctly classified by the ResNet (with threshold 0.96, see § 3.2.2.4). The orange histogram shows the number of lenses with a U-Net detection threshold above 0.5, and the gr…
Figure 29
Figure 29. Figure 29: Hexbin plot of the distance between the ground truth and U-Net prediction versus the brightest pixel to ground truth flux ratio. The top panel (entire validation set) includes predictions on all images in the validation set, the middle panel (only ResNet) includes pre…
Figure 30
Figure 30. Figure 30: Examples of lenses that are correctly classified and detected by the ResNet and U-Net models, respectively. All images have a ResNet classification probability above 0.96 and U-Net detection probability above 0.72, where the RUN pipeline achieves a pixel level precisi…
Figure 31
Figure 31. Figure 31: A relatively straightforward image for lens detection by the U-Net. The quantity “Distance” at the top left of the figure indicates the separation between the predicted lens location and the true lens location in pixels. The probability heat map (lower left) shows tha…
Figure 32
Figure 32. Figure 32: A challenging lens that the U-Net detects correctly. There are many objects around the lens that make it difficult to detect the correct location of the lens. The U-Net nevertheless correctly predicts the lens location. This challenging example provides promising evid…
Figure 33
Figure 33. Figure 33: A lens with an Einstein radius of θE = 0.03′′. Even though the lens is not the brightest pixel in the image and has a very small Einstein radius, the U-Net is able to confidently detect the lens with a 0.7243 detection probability [PITH_FULL_IMAGE:figures/full_fig_p0…
Figure 34
Figure 34. Figure 34: Even though the prediction (red cross hairs, barely visible) is one pixel away from the ground truth (white cross hairs), the detection is considered as correct (see Section 4.2.1), as the model would still identify the relevant location in the image for human inspect…
Figure 35
Figure 35. Figure 35: Trained Model 1b (HST-short) tested on single-exposure lenses (top two rows) and non-lenses (bottom two rows). From HST-GO 10174, with exposure time of 420s. The probability the model predicted is shown for each image. Here we emphasize that a) all images in this test…
Figure 36
Figure 36. Figure 36: Example of the lensing effect of a lens with Einstein radius θE = 0.032′′, shown using a standard SIS and a truncated SIS lens model [PITH_FULL_IMAGE:figures/full_fig_p064_36.png]
Figure 37
Figure 37. Figure 37: Precision and recall for Model 3 (JWST-small) as a function of probability threshold. The solid lines show the performance on the original validation set with SIE lenses, and the dotted lines show the performance on the validation set with truncated SIS lenses. We gen…
Figure 38
Figure 38. Figure 38: Precision-Recall curve for the ResNet and U-Net pipeline, evaluated on the SIS and truncated SIS simulations. Colored shapes represent U-Net probability thresholds that are multiples of 0.1 (leftmost is 0.9), corresponding to the curve of the same color. Across most p…
Figure 39
Figure 39. Figure 39: Histogram of the VELA galaxies used in the training (blue bars) and validation (orange bars) datasets for the k-fold 0th split. VELA images of the galaxies at z = 1 with a randomly chosen angle are shown above the histogram. expected; but if its performance is signifi…
Figure 40
Figure 40. Figure 40: The AUC performance of each k-fold model evaluated on its corresponding validation set compared to the baseline model’s performance (from § 3.2.2.2) for the ResNet model. The horizontal axis indicates the validation set used [PITH_FULL_IMAGE:figures/full_fig_p068_40.png]
Figure 41
Figure 41. Figure 41: The AUC performance of each k-fold model evaluated on its corresponding validation set compared to the baseline model’s performance (from § 3.2.2.2) for the U-Net model. The horizontal axis indicates the validation set used. Note that the AUC is calculated using the p…
Figure 42
Figure 42. Figure 42: While training each k-fold model for [PITH_FULL_IMAGE:figures/full_fig_p069_42.png]
Figure 43
Figure 43. Figure 43: The AUC performance of the model trained with only S´ersic environmental galaxies compared to the baseline model (from § 3.2.2.2). F. LENS DETECTION EXAMPLES First we want to make it clear that in this section all three systems shown have a lower U-Net detection proba…
Figure 44
Figure 44. Figure 44: A non-lens that is classified as a lens by the ResNet and U-Net model (false positive). We note that for this system, the predicted probabilities are below the chosen thresholds of 0.96 and 0.72 for ResNet and U-Net, respectively. There is a very bright environmental …
Figure 45
Figure 45. Figure 45: A lens that is correctly classified by the ResNet (with a probability higher than the chosen threshold of 0.96), but incorrectly detected by the U-Net (of course, with a probability lower than the chosen threshold of 0.72). Since an environmental galaxy in the image i…
Figure 46
Figure 46. Figure 46: A lens that is correctly classified by the ResNet (with a probability higher than the chosen threshold of 0.96), but incorrectly detected by the U-Net (again, with a probability lower than the chosen threshold of 0.72). In this case, the model’s prediction (red crossh…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.