Surrogate modeling of gravitational waves microlensed by spherically symmetric potentials

Gopalkrishna Prabhu; Md Arif Shaikh; Scott E. Field; Shasvath J. Kapadia; Uddeepta Deka; Vijay Varma

REVIEW 3 major objections 7 minor 1 cited by

Surrogate modeling of gravitational waves microlensed by spherically symmetric potentials

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Surrogate modeling reproduces microlensed gravitational-wave amplification factors for point-mass and singular isothermal sphere lenses, with mismatches below $5\times10^{-4}$ and evaluation near 100 ms.

desk verdict A solid proof-of-principle that surrogates can speed up evaluation of the time-domain microlensing amplification factor; the main caveat is that the training truth itself is a reconstructed quantity, though it is anchored to analytic PML and SIS results. read the letter →

arxiv 2501.02974 v3 pith:DOKAGGXV submitted 2025-01-06 gr-qc

Uddeepta Deka , Gopalkrishna Prabhu , Md Arif Shaikh , Shasvath J. Kapadia , Vijay Varma , Scott E. Field This is my paper · ORCID

classification gr-qc

keywords gravitationalwavemicrolensingsurrogatemodelingamplificationfactoropticstime-delaycurvespoint-masslenssingularisothermalspherereducedbasis

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

Gravitational-wave microlensing imprints wave-optics features on a signal through an amplification factor, but most realistic lenses have no closed-form expression for it, and numerical evaluation is too slow for the large Bayesian searches that a detection will require. This paper establishes that surrogate modeling, an interpolation method already used for binary-black-hole waveforms, can instead be applied to the time-domain amplification factor $\widetilde{F}(t)$, which is smooth enough to interpolate after the logarithmic saddle-image peak is reconstructed analytically and then regularized. Building surrogates for the point-mass lens and the singular isothermal sphere, the paper reports microlensed waveforms that match the underlying numerical lens models with mismatches below about $5\times10^{-4}$ and evaluate in about 100 ms, between 5 and 1000 times faster than the original models. If this holds, parameter estimation of microlensed gravitational-wave events becomes practical rather than prohibitive.

What carries the argument

Two constructions do the work. First, the amplification factor is computed and modeled in the time domain: $\widetilde F(t)=dS/dt$, the rate of change of the area on the lens plane between curves of constant time delay, which avoids the rapidly oscillating Kirchhoff integrand. Second, the surrogate itself is a reduced-basis and empirical-interpolation model: a greedy algorithm selects a small set of representative waveforms, empirical interpolation chooses sparse time nodes at which the waveform must be known, and Gaussian-process regressions fit the dependence on the impact parameter $y$ at those nodes. Two preprocessing steps make interpolation accurate near the saddle-image peak: the peak is reconstructed by splicing a rescaled analytic logarithmic approximation over an empirically chosen $\pm5\%$ window around $t_{\mathrm{peak}}$, and the regularization $\bar F(t)=1-\exp[-\pi(\widetilde F(t)-1)/\sqrt{\mu_-}]$ removes the logarithmic divergence so the interpolation target is smooth. The lens mass $M_L$ only sets the time scale, so each model is a single-parameter surrogate in $y$.

What would settle it

Take the point-mass-lens surrogate and compare its frequency-domain amplification factor directly with the exact analytic expression of Eq. (A6) at the upper end of the next-generation detector band (for example $M_L=10^3\,M_\odot$ and $y=0.1$), instead of comparing with the reconstructed numerical reference. If the mismatch there exceeds the reported $\sim5\times10^{-4}$, the peak-reconstructed training reference is biased at high dimensionless frequency and the surrogate's advertised accuracy is not the true error. The same check can be run for the SIS surrogate against the high-order summation form of Eq. (A13).

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Extended reading notes

Core claim

The paper's central claim is that the time-domain amplification factor $\widetilde F(t)$, defined by $\widetilde F(t)=\int d^2x\,\delta(\tau_d(x,y)-t)$ and computed as the area swept out between neighboring constant-time-delay curves, is a suitable surrogate-modeling target even though the frequency-domain factor $F(w)$ is highly oscillatory. For a point-mass lens over impact parameters $y\in[0.1,2]$ and a singular isothermal sphere over $y\in[0.01,0.95]$ (the two-image regime), the authors first reconstruct the logarithmic peak contributed by the saddle image using the analytic near-peak form of Ref. [60], then apply the regularization $\bar F=1-\exp[-\pi(\widetilde F-1)/\sqrt{\mu_-}]$, and build reduced-basis surrogates with 27 and 15 basis vectors at a basis tolerance of $5\times10^{-4}$. After inverse regularizing and Fourier transforming, the surrogate amplification factor is multiplied into unlensed binary-black-hole waveforms; the resulting mismatches against the numerical reference are below $5\times10^{-4}$ across the tested $M_L$-$y$ grid and for binary total masses from $20$ to $100\,M_\odot$, with frequency-domain waveform evaluation times of order $10^{-2}$-$10^{-1}$ s. The claim is that this makes surrogate modeling a viable route to fast, accurate microlensed templates in regimes where no analytic amplification factor exists.

Load-bearing premise

The reference waveforms used to train and test the surrogate are not direct solutions of the Kirchhoff diffraction integral: they come from the area-between-time-delay-curves formula of Eq. (7), with the logarithmic peak replaced by a rescaled analytic approximation over an empirically chosen $\pm5\%$ matching window. If that reconstructed reference is biased at high frequencies, or if the 5% window is not robust across parameter space, the reported mismatches understate the surrogate's true error against the actual lensed waveform.

Editorial extensions

If this is right

Large-scale Bayesian parameter estimation of microlensed gravitational-wave signals becomes computationally feasible, since surrogate templates evaluate in about 100 ms rather than seconds.
The same time-domain prescription, compute $\widetilde F(t)$, reconstruct and regularize the peak, and build a reduced-basis surrogate, extends to any lensing potential for which numerical time-delay solutions can be produced, without needing an analytic $F(f)$.
For the point-mass lens the surrogate is faster than the existing lookup-table interpolation used in current searches while remaining fully wave-optics based.
The singular-isothermal-sphere surrogate demonstrates that a logarithmic divergence in $\widetilde F(t)$ does not block accurate interpolation when the peak is handled analytically before surrogate construction.

Reading between the lines

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

A direct high-frequency validation against the analytic point-mass result, rather than the reconstructed reference, would sharpen the accuracy claim, since the paper's own discussion identifies peak resolution as the main high-frequency error source.
Because the lens mass only rescales time, the single-parameter-in-$y$ surrogates can be extended to a two-parameter $(M_L,y)$ model simply by rescaling the time axis, which would make them directly usable in parameter estimation.
For asymmetric or multi-lens configurations the paper notes that caustics split parameter space into regions with different image counts; separate surrogates per region are a plausible path, but the caustic geometry itself would need to be modeled.
The rate-determining step is the Fourier transform, not the surrogate evaluation, so faster FFT algorithms or direct frequency-domain interpolation would push the total cost well below 100 ms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: this is a clean, incremental methods paper. It builds reduced-basis/GPR surrogates for the time-domain lensing amplification factor Ftilde(t) for a point mass and a singular isothermal sphere, reports mismatches below 5e-4 against the numerical reference, and is 5–1000x faster. The speed claim holds up. The novelty is applying surrogate modeling to the amplification factor itself, in the time domain, rather than to the final waveform; that is a real and useful step, not a conceptual breakthrough.

What the paper does well: the validation is honest. Training and test parameters are disjoint, the match is a standard detector-weighted overlap, and the greedy convergence curves plateau. They benchmark the PML surrogate against the LVK lookup table and show it is faster. They are also upfront about the caustic/multiple-image limitation, which sets the right scope for a proof-of-principle. The numerical reference is calibrated to analytic results (closed-form PML, SIS series truncated at n=500) at a claimed mismatch below 1e-4, so the surrogate is not purely self-referential.

The soft spots are real but not fatal. The main one is the reference truth. The numerical Ftilde(t) is built from the area-between-time-delay-curves expression plus a 5% window splice of the analytic logarithmic peak. The 5% is empirically chosen, and there is no sensitivity study showing that the window width is robust across impact parameter and lens mass. For the SIS series, the n=500 truncation has no convergence check; the 1e-4 calibration statement is asserted, not demonstrated. If the reference carries a systematic error near the peak or at high frequencies, the reported mismatches could understate the true error against the actual lensed waveform. I think that risk is moderate, not disqualifying, because the analytic anchors keep the reference from being wildly wrong, but it should be shown rather than asserted. Second, no code or data is shipped. For a methods paper that is a genuine gap; the training data and evaluation scripts would make the work reproducible and much more useful. Third, the SIS surrogate is compared only against the numerical solver, not against the existing time-domain lookup table (Ref 95). That is the closer baseline and its absence weakens the speed claim.

Who this is for: anyone working on microlensed GW parameter estimation with next-generation detectors. It is a practical tool paper, not a physics breakthrough.

My recommendation: send it out for peer review. The referee should ask for the code and data, a convergence check on the SIS series, and a short study varying the 5% matching window. With those, this becomes a solid methods contribution.

Referee Report

3 major / 7 minor

Summary. The paper proposes constructing surrogate models of the time-domain gravitational-wave microlensing amplification factor \tilde F(t) for two spherically symmetric lens models, the point-mass lens (PML) and the singular isothermal sphere (SIS), and then obtaining fast microlensed waveforms by Fourier transforming the surrogate amplification factor and multiplying by an unlensed frequency-domain waveform. The training data are generated from the area-between-time-delay-curves expression of Eq. (7), with the logarithmic peak reconstructed from the analytic near-peak formula Eq. (12) inside an empirically chosen 5% matching window, followed by the regularization of Eq. (13). The surrogate itself uses reduced-basis greedy selection, empirical interpolation, and Gaussian process regression. The authors report mismatches \lesssim 5 \times 10^{-4} against the numerical microlensed waveforms across the tested ML-y grids, evaluation times of about 100 ms, and speedups between 5 and 10^3 relative to the underlying lensing models.

Significance. If the accuracy claim holds, the paper supplies a practically useful ingredient for parameter estimation of microlensed gravitational waves, and the methodology is in principle extendable to more complex, asymmetric lens models. The internal validation is careful in several respects: training and test lens parameters are disjoint, mismatches are computed with a standard detector-weighted match using the Einstein Telescope PSD, the greedy convergence curves in Fig. 5 plateau, and speed is measured against a lookup table for PML and against direct numerics for SIS. The main unresolved risk is that the validation truth is itself a reconstructed numerical reference; the manuscript does not supply a calibration error map, an SIS series-convergence check, or a sensitivity study of the empirical 5% reconstruction window. These omissions directly affect the central accuracy claim rather than the speed claim.

major comments (3)

[Section II B and Section III] The central accuracy claim is validated only against a training/validation truth that is itself reconstructed: \tilde F(t) from Eq. (7) is not a direct Kirchhoff-integral solution, and the logarithmic peak is replaced by the analytic approximation Eq. (12) inside an empirically chosen 5% matching window. Section III states that the numerical amplification factors are calibrated to the analytic PML form (Eq. A6) and the SIS series (Eq. A13) so that the maximum mismatch is < 10^{-4}, but no calibration error map, no convergence study for the n=500 SIS series truncation, and no sensitivity of the 5% window width are shown. If the reconstruction bias is larger than 10^{-4} at the high dimensionless frequencies sampled at the largest lens masses, the reported surrogate mismatches, which are also computed against this reconstructed reference, would understate the error relative to the true lensed waveform. Please add a calibration error map for both lens models, a convergence test for the SIS series, and a variation of the matching window width (for example 2% and 10%) with the resulting mismatch values.
[Section III, Fig. 7] Because the PML has the closed-form F(w) in Eq. (A6), the final surrogate should be compared directly against this analytic solution, not only against the calibrated numerical pipeline. Such a comparison would remove the reconstructed-reference issue for the PML entirely and would directly support the headline claim that the surrogate reproduces the original lens model. As it stands, Figs. 7 and 8 only compare against the numerical method that was used to construct the surrogate, so the validation does not independently confirm the accuracy claim for the one model where an exact reference is available.
[Section II D 3 and Fig. 5] The greedy search is stopped when the normalized L2 error in the regularized time domain is below 10^{-2}, while the basis tolerance is 5 \times 10^{-4}, but the relation between these thresholds and the advertised frequency-domain mismatches below 5 \times 10^{-4} is not established. The final mismatches are computed separately, so this is not an internal contradiction, but the manuscript should clarify whether the reported mismatches are sensitive to the L2 stopping criterion and the basis tolerance; a small variation of these two parameters with the resulting mismatch statistics would make the empirical choices in the greedy loop less opaque.

minor comments (7)

[Eq. (13)] The symbol \mu_- is signed negative for the saddle image in Eq. (A8), but Eq. (13) takes its square root; the formula should use |\mu_-|, or \mu_- should be redefined as the absolute magnification.
[Eqs. (15) and (20)] The summation indices in Eqs. (15) and (20) appear as 'i=i' and 'j=i'; these should be 'i=1' and 'j=1'.
[Eq. (23)] The set C in min_{c_i \in C} is not defined; presumably it is \mathbb{R}^m, and this should be stated for clarity.
[Section III] The relationship between the number of basis vectors (27 for PML, 15 for SIS) and the number of greedy training parameters (35 for PML, 100 for SIS) should be clarified; the text is easy to misread as saying the basis is built from all 35 or 100 greedy points.
[Abstract and Section IV] The abstract states mismatches \lesssim 5 \times 10^{-4}, the introduction mentions O(10^{-7}-10^{-3}), and Section IV reports 10^{-8}-10^{-4}; these ranges should be made consistent with the summary statistics behind Fig. 8.
[Fig. 8, top-right panel] The histogram of evaluation times is described as comparing the surrogate with the lookup table, but the caption does not state whether the surrogate time includes the FFT that the text identifies as the rate-limiting step; please specify.
[General] A data and code availability statement would improve reproducibility, since the numerical method and the peak-reconstruction procedure involve several implementation choices that are only described at a high level.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the surrogate's accuracy is benchmarked against independently calibrated numerical references, not against its own training outputs.

full rationale

The paper's central claim is that a surrogate model can rapidly and accurately reproduce the time-domain microlensing amplification factor. The surrogate is an interpolant, so it is by construction close to the training set; however, the reported accuracy is evaluated at lens parameters different from those used to construct the surrogate, as stated: 'We ensure that the lensing parameters used for training the model, and those used for testing (i.e, evaluating the mismatches), are different.' The mismatches are therefore generalization errors, not identities. The numerical reference used for training and validation is independently anchored: the paper states that 'the numerical amplification factors used to train the model for PML and SIS are calibrated to the analytic form given in Eq. (A6) for PML and the summation form given in Eq. (A13) for SIS, such that the maximum mismatch (1−M) between the numerically computed lensed waveform and the analytically obtained lensed waveform is < 10−4 for all possible points in the parameter space.' The analytic PML expression and the SIS series are external results (Peters 1974; Matsunaga and Yamamoto 2006), not outputs of this paper. The peak reconstruction uses the analytic logarithmic approximation of Eq. (12) from Ulmer and Goodman, with image magnifications and time delays from the lens models themselves; the 5% matching window is empirical but it is a preprocessing choice, not a fitted parameter used to force the final mismatches. The paper does cite prior work with overlapping authorship for the area-between-time-delay-curves numerical method ([51]) and for surrogate modeling techniques ([61], [62], [69]), but these citations are methodological, not load-bearing for the validation claim: the accuracy chain terminates in externally known analytic or series solutions. The empirically tuned peak-matching window and the truncated SIS series are legitimate accuracy concerns, but they are correctness risks about the reference truth, not circularity. No step in the derivation defines the predicted quantity in terms of the fitted result or imports a conclusion exclusively from a self-citation. The result is a standard interpolation benchmark with independent anchoring, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the numerical reference construction (area method plus peak reconstruction), on standard lensing assumptions, and on several empirically chosen hyperparameters. No new physical entities are introduced. The surrogate itself is an emulator, not a physical model.

free parameters (3)

peak reconstruction window = 5% of tpeak on each side
Section II B(d): window span empirically chosen; larger span reduces accuracy of Eq. (12), smaller depends on numerical accuracy.
reduced basis tolerance = 5e-4
Section II D3: chosen so surrogate achieves required accuracy without overfitting; affects number of basis vectors and evaluation speed.
greedy L2 stopping threshold = 1e-2
Section II D3: greedy search stops when maximum L2 error below 1e-2; controls training set size and final accuracy.

assumptions (4)

domain assumption Thin-lens approximation and scalar Kirchhoff diffraction integral (Eq. 5) describe microlensed GW propagation.
Standard in lensing literature; invoked in Section II A without independent derivation.
domain assumption The time-domain factor Ftilde(t) can be obtained as area between time-delay contours (Eq. 7) following Ulmer and Goodman.
The numerical solver relies on this identification; no independent numerical cross-check is provided.
domain assumption The logarithmic peak behavior near the saddle image is given by Eq. (12) with magnifications and tpeak from geometric optics (Eqs. A8/A9 for PML, A15/A16 for SIS).
Used in the peak reconstruction (Section II B); not derived in the paper.
domain assumption The SIS lens is restricted to y<1 so exactly two images exist; no caustic crossing is modeled.
Stated in Section III and Section IV as a limitation.

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Cite this review

Pith. "Pith review of Surrogate modeling of gravitational waves microlensed by spherically symmetric potentials." pith.science (2026). https://pith.science/paper/DOKAGGXV

@misc{pith2026250102974,
  author       = {Pith},
  title        = {Pith review of: Surrogate modeling of gravitational waves microlensed by spherically symmetric potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DOKAGGXV}},
  note         = {Machine review of arXiv:2501.02974}
}

abstract

The anticipated observation of the gravitational microlensing of gravitational waves (GWs) promises to shed light on a host of astrophysical and cosmological questions. However, extracting the parameters of the lens from the modulated GWs requires accurate modeling of the lensing amplification factor, accounting for wave-optics effects. Analytic solutions to the lens equation have not been found to date, except for a handful of simplistic lens models. While numerical solutions to this equation have been developed, the time and computational resources required to evaluate the amplification factor numerically make large-scale parameter estimation of the lens (and source) parameters prohibitive. On the other hand, surrogate modeling of GWs has proven to be a powerful tool to accurately, and rapidly, produce GW templates at arbitrary points in parameter space, interpolating from a finite set of available waveforms at discrete parameter values. In this work, we demonstrate that surrogate modeling can also effectively be applied to the evaluation of the time-domain microlensing amplification factor $\widetilde{F}(t)$. We show this by constructing $\widetilde{F}(t)$ for two lens models, viz. point-mass lens, and singular isothermal sphere, which notably includes logarithmic divergence behaviour. We find both surrogates reproduce the original lens models accurately, with mismatches $\lesssim 5 \times 10^{-4}$ across a range of plausible microlensed binary black hole sources observed by the Einstein Telescope. This surrogate is between 5 and $10^3$ times faster than the underlying lensing models, and can be evaluated in about 100 ms. The accuracy and efficiency attained by our surrogate models will enable practical parameter estimation analyses of microlensed GWs.

Figures

Figures reproduced from arXiv: 2501.02974 by the authors.

**Figure 2.** Top row: The time-domain amplification factor, Fe(t), and its regularized version defined in Eq. (13), eF(t), due to a point-mass lens for various impact parameters y as a function of time t (in units of 4GML(1 + zL)/c 3 , where the global minima lies at t = 0). Bottom row: Amplitude of the lensed waveforms (left) considering a GW150914-like source and the amplitude and phase of the frequency-domain amplification fa… view at source ↗

**Figure 3.** Fe(t) for various impact parameters y for the SIS lens model computed using the numerical method (solid lines) and the surrogate model (dashed lines) as a function of time t (in units of 4GML(1 + zL)/c 3 , where the global minima lies at t = 0) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Illustration of improvement in the accuracy of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Largest relative L2 error of the surrogate for the PML model (top) and the SIS model (bottom) as a function of number of greedy parameters. The error is computed as the maximum error between the entire validation set and each surrogate model. Both surrogate models are …

**Figure 7.** Figure 7: shows the mismatch between the surrogate microlensed waveform and the microlensed waveform obtained using the numerical method for a GW150914-like signal for various values of ML and y. The unlensed waveform is gen10−1 100 y 200 400 600 800 1000 ML −7 −6 −5 −4 log10(…

**Figure 6.** Figure 6: Evaluation time for the amplification factors due to point [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 8.** Figure 8: Top panel: (Left) Histograms of mismatch between the lensed waveforms due to PML computed using the surrogate model and the numerical method. Each histogram corresponds to a different source with the total source mass, Mtot = {20, 40, 60, 80, 100}M⊙. The mismatches are…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

II and III of Ref

Surrogate modeling basics We follow the methods described in Sec. II and III of Ref. [61] and refer the reader to it for a detailed overview of building a surrogate model. In the following, we briefly outline building a surrogate model

[2] [2]

Given a known set of solutions (also called the training set), the RB is found using a greedy search algorithm

The first step is to find a minimal set of solutions called the reduced basis (RB) [84] in terms of which solutions at other parameters can be expressed. Given a known set of solutions (also called the training set), the RB is found using a greedy search algorithm. For example, with a RB with m basis elements,{ei}m i=1, a solution eF(t; λ) in the training...

[3] [3]

The next step is to find the most representative times, the empirical times or nodes{Ti}m i=1 to construct an in- terpolant in time using empirical interpolant method (EIM) [61, 85–88] for a given parameter λ: Im[eF](t; λ) = mX i=i Ci(λ) ei(t). (15) The coefficients Ci(λ) are defined by requiring that the interpolant becomes equal to the value of the solu...

[4] [4]

(20) at hand, we need to know the values of the solutions at the empirical nodes{Ti}m i=1

With the interpolantIm[eF](t; λ) in Eq. (20) at hand, we need to know the values of the solutions at the empirical nodes{Ti}m i=1. Therefore, the third step is to construct parametric fits at each empirical node across the param- eter space

[5] [5]

Finally, the surrogate model prediction at a new parame- ter λ∗ outside the training data set is computed by first using the parametric fits (constructed in the third step) to evaluate the values ofeF(t; λ∗) at the empirical nodes {Ti}m i=1 and then building the interpolantIm[eF](t; λ) us- ing Eq. (20). The first three steps are performed offline (i.e, pr...

[6] [6]

How- ever, since the effect of microlensing is a frequency-dependent modulation of the GW amplitude as explained in Sec

Surrogate for microlensed waveforms We apply the technique of surrogate modeling to obtain a model of the GWs waveform undergoing microlensing. How- ever, since the effect of microlensing is a frequency-dependent modulation of the GW amplitude as explained in Sec. II A, one needs only to model this amplification factor provided a fast, unlensed model exis...

[7] [7]

lookup table

Surrogate modeling setup The RB is found by using a greedy algorithm that guarantees that the error σm≡ max λ min ci∈C eF(.; λ)− mX i=1 ci(λ) ei(.) , (23) associated with the approximation in Eq. (14) (minimized over the coefficients ci and maximized over the parameter λ) is less than a desired value, usually called the basis tolerance. Here, we have defi...

2023

[8] [8]

Advanced LIGO,

J. Aasi et al. (LIGO Scientific), “Advanced LIGO,” Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc]

arXiv 2015