Identifiability Limits of Gravitational Wave Phase Deviations: Multiclass Classification with a Multihead Neural Network

• The network classifies GR versus six ppE phase families and regresses the logarithm of the coupling magnitude from the mismatch response function.The families use exponents b ∈ {−7, −5, −3, −1, +1, +2}.Source: §1 · p. 3
• A phase-family label identifies frequency dependence, not necessarily a unique gravity theory, because distinct theories can share an exponent.Identifying a specific theory requires additional information, such as predictions for further phase terms.Source: §1 · p. 3
• The analysis fixes GR-template intrinsic parameters to injected values, making its setup an optimistic benchmark rather than an upper bound for data-based parameter estimation.Fitting intrinsic parameters could absorb deviation components, but the resulting change in classification accuracy cannot be inferred from residual norm alone.Source: §1 · p. 3
• The source-conditioned residual reference helps distinguish network errors from confusion arising from similarities among phase families.The paper also examines local geometry, fixed-false-positive detection, auxiliary regression, recorded Hanford strain, and network variants.Source: §1 · p. 3
• Related methods address modified-phase searches, detection of PN deviations, recorded-noise waveform modulations, or continuous latent representations rather than this discrete-family classification setup.This network produces classification scores and an auxiliary point estimate, not joint source-and-deviation posteriors.Source: §1 · p. 4

2.1 Signals, catalog and detectors

• The datasets use 262 catalog sources at Advanced LIGO design sensitivity and 132 at Hanford O3 sensitivity, with the Hanford set meeting the stated source-selection requirements.Each source is evaluated in 100 independent stationary Gaussian-noise realizations; the Advanced LIGO and Hanford cutoffs are 10 Hz and 20 Hz.Source: §2.1 · p. 4

2.2 Deviations

• The six phase families are indexed by ppE exponent b, with examples of theories in the family table; several theories share an exponent.The analysis retains phase corrections, tapers them to the inspiral, and uses the modified waveform without linearizing its deviation.Source: Table 1 · p. 5, §2.2 · pp. 5–6
• The taper is applied to the phase correction’s second frequency derivative, then integrated twice to recover a smooth phase that vanishes above the taper.This avoids a spurious curvature contribution from directly windowing the phase; b = 3 is excluded because its phase term is linear in frequency and has zero second derivative.Source: §2.2 · p. 6

2.3 Template and response function

• The mismatch response function is the frequency kernel for the first-order change in mismatch under a phase deformation, and the network receives its real and imaginary components.The GR template keeps injected intrinsic parameters while amplitude and arrival time are determined from each noisy realization; input-channel normalization discards separate channel scales.Source: §2.3 · pp. 7–8

2.4 Sampling protocol

• Samples are drawn using residual norm s, with a log-uniform strength range and limits that keep deviations small, avoid over-weighting the loudest sources, and preserve a unique coupling magnitude.Each source and noise realization contributes one GR sample and one per exponent with shared noise; sources are retained only when all six deviations are visible in band, with Hanford also requiring SNR ≥ 8.Source: §2.4 · pp. 8–9
• The reported strength-dependent results use sexact, the residual norm actually present after maximizing the full time shift, rather than the linearized sampling coordinate s.The datasets contain 183,400 Advanced LIGO samples and 92,400 Hanford samples, with source-separated training, validation, and test splits.Source: §2.4 · p. 9

3.1 Network architecture and training

• The network processes the response function with convolutional blocks and a bidirectional GRU, then uses shared features for class probabilities and standardized log10 |β| regression.The architecture also includes two output heads: a seven-class softmax and a linear regression output.Source: §3.1 · p. 9, Figure 1 · p. 10

3.2 Reference classifier

• The residual bank tabulates noise-free post-match residuals by source, exponent, sign, and strength, then scores observed residuals against those templates and GR.It interpolates between 48 logarithmically spaced strengths for each source, exponent, and sign.Source: §3.2 · pp. 10–11
• Because time matching depends on noise, the residual-bank likelihood is approximate, and its source-specific absolute-residual information exceeds the network's normalized input, so it is not a strict upper bound.The reference also uses known source parameters, whereas the network input's channel normalization removes absolute residual size.Source: §3.2 · p. 11
• The section also compares the residual bank with a linear-ray reference for nonlinear residual effects and with PCA followed by LDA as a linear classifier.The linear-ray reference replaces exact residuals with small-coupling residual directions.Source: §3.2 · p. 11

3.3 Metrics and uncertainty

• Metrics include seven-class and loud-sample accuracy, GR recall, deviation-search efficiency and identification at a 1% validation GR false-positive threshold, and coarse adjacent-exponent summaries.Loud samples have s_exact > 5; efficiency counts flagged beyond-GR samples, while identification counts those flagged with the correct exponent.Source: §3.3 · pp. 11–12
• Uncertainties are estimated by bootstrapping sources 10,000 times, with recorded-noise segments redrawn and classifier comparisons evaluated on matched draws.For the network, decisions from three seeds are averaged for each sample before comparison.Source: §3.3 · pp. 11–12

4 Geometry of the deviations

• At small coupling, each family defines a residual direction after the match; small angles between directions mean strong deviations are needed to distinguish the families.Because either coupling sign is allowed, near-parallel and near-antiparallel residuals both correspond to small angles.Source: §4 · p. 12
• All six phase-family residual directions lie within about 25 degrees, neighboring exponents differ by only a few degrees, and the closest pair is b = +1 and b = +2 for every source.The smallest angle decreases with source mass and is smaller on Hanford, where the lower cutoff shortens the observed frequency band.Source: §4 · p. 12, Figure 2 · p. 13
• For loud samples, the exact residual bank identifies the exponent more often than the linear-ray reference because strong-deviation residuals bend away from their linear directions.The linear-ray reference uses the small-coupling directions, while the residual bank retains nonlinear residual structure.Source: §4 · p. 12

5.1 Classification and comparison with the reference

• 0.427 and 0.329 are the network’s loud-sample exponent accuracies on aLIGO and Hanford, versus 0.469 and 0.405 for the residual bank.Across all samples, the network is about one percentage point below the residual bank on both rows; the Hanford loud subset contains only four sources.Source: §5.1 · p. 13, Table 2 · p. 14
• Accuracy rises with deviation strength, while on loud aLIGO samples the residual bank and network identify exponents more often for lighter than heavier sources.The bank reaches 61% below 30 M⊙ and 37% above 60 M⊙; the network reaches 53% and 34%, respectively. About 90% of network errors also occur among the bank’s errors.Source: §5.1 · pp. 13–14, Figure 3 · p. 14

5.2 Confusion and coarse groups

• Most loud-sample errors fall on neighboring exponents, and merging classes into GR, negative exponents, and positive exponents yields about 80% correct grouping on aLIGO.The outer families are identified in more than 60% of loud aLIGO samples, while roughly a third of wrong loud decisions there fall beyond neighboring exponents.Source: §5.2 · p. 15, Figure 4 · p. 15, Table 3 · p. 16

5.3 Detection and calibration

• Nearly every loud deviation is flagged, but fewer than half receive the correct exponent; class probabilities have expected calibration error of about one percent or less.The detection threshold is set to flag 1% of GR validation samples, and the GR test false-positive fraction stays close to 1%.Source: §5.3 · pp. 16–17, Table 4 · p. 17

5.4 The auxiliary regression head

• The regression head’s family-average predictions are pulled toward the center of the coupling range and are worse than using each family’s average coupling.It overestimates small b = −7 couplings and underestimates large b = +2 couplings by about three orders of magnitude on average; channel normalization removes response-function scale.Source: §5.4 · pp. 17–18, Figure 6 · p. 18

5.5 PCA and a linear baseline

• LDA separates families weakly, reaching 0.20 rather than the network’s 0.43 accuracy on loud aLIGO samples, while PCA projections separate deviations by phase-curvature sign rather than family.The curvature groups mix the sign of the coupling with the sign of the exponent; the network uses response-function structure the linear classifier does not capture.Source: §5.5 · pp. 18–19, Figure 7 · p. 19

6 Injections into recorded Hanford O3 strain

• The test uses selected 64-second Hanford O3 stretches after excluding known events, flagged times, and stretches with excess noise power.Noise spectra are estimated from surrounding 1024-second intervals using the median Welch method.Source: §6 · p. 19
• The recorded-noise test uses O3b data not seen during training, while training and validation use O3a.The recorded-noise dataset is otherwise identical to the Gaussian Hanford dataset, enabling sample-by-sample comparison.Source: §6 · p. 19
• Accuracies hardly change between Gaussian and recorded noise; on recorded O3b, loud accuracies differ by at most about two percentage points.The networks reach almost the same overall accuracy on recorded O3b as on Gaussian noise.Source: §6 · p. 19, Table 5 · p. 20
• On recorded O3b, the false-positive fraction is 1.6% for Gaussian-trained networks and 1.1% for recorded-noise-trained networks.The difference is not yet significant with the limited test data.Source: §6 · pp. 19–20, Table 5 · p. 20

7 Exploration of network variants

• None of the thirteen variants exceeds the retrained recurrent network in loud accuracy, and all remain below the residual bank.Recurrent versions come closest; most pooled variants are lower by about two to four percentage points.Source: §7 · pp. 20–21, Table 6 · p. 21
• Recurrent read-out preserves frequency order, matching the frequency-dependent pattern that distinguishes the six phase families.Pooled variants average each feature across the band; those given extra information such as frequency position come closer to recurrent performance, while wider or deeper pooled networks do not help.Source: §7 · p. 21
• The comparison uses only two seeds per variant, and most variants change several design choices at once.For example, the classification-only variant changes both read-out and regression loss, so it cannot isolate the effect of the regression loss.Source: §7 · p. 21
• The tested variants indicate that recurrent networks are a good choice, but do not establish which individual ingredient matters most.Small differences should not be over-interpreted given the seed count and comparison design.Source: §7 · pp. 21–22

8 Conclusions and outlook

• The analysis uses 262 sources at aLIGO design sensitivity and 132 at Hanford O3 sensitivity, with intrinsic source parameters fixed at injected values.Signals were evaluated in Gaussian noise and recorded Hanford strain.Source: §8 · p. 22
• The network identifies loud-deviation exponents in 43% of aLIGO cases and 33% of Hanford cases, close to the approximate residual bank.The network and residual bank make largely the same errors, which follow from neighboring exponents leaving nearly parallel residuals after matching.Source: §8 · p. 22
• A loud deviation is flagged in nearly every case, about 1% of GR signals are flagged, and the sign is identified in about 80% of loud aLIGO samples.The principal difficulty is exact exponent identification rather than deviation detection or rough frequency dependence.Source: §8 · p. 22
• An initial analysis finds loud-sample exponent accuracy of 0.63 for Einstein Telescope, 0.48 for Cosmic Explorer, and 0.43 for aLIGO.The authors attribute improvement partly to stronger signals and, for Einstein Telescope, sensitivity extending to lower frequencies.Source: §8 · p. 22
• The regression head does not estimate coupling reliably, and the analysis’s main restriction is using true intrinsic source parameters.Estimating those parameters is expected to absorb part of each deviation and lower accuracy; higher modes, precession, detector networks, and controlled architecture comparisons remain future extensions.Source: §8 · pp. 22–23

A Derivation of the mismatch response function

• The derivation holds data, spectrum, integration band, and injected-source intrinsic parameters fixed while deriving the mismatch response from the residual.The residual is defined as r = d − hm.Source: A Derivation of the mismatch response… · p. 23
• The match is maximized over arrival time and template phase, and the template is aligned so its complex overlap with the data is real and positive.At the maximizing time and phase, first-order changes in alignment do not change the match, allowing the template to be varied with alignment held fixed.Source: A Derivation of the mismatch response… · pp. 23–24
• A small phase perturbation is represented as h → h exp[iϵξ(f)], where ξ(f) is a real profile and ϵ is a small real parameter.Because a phase change preserves template norm, the term proportional to the match contributes only to the imaginary part of the response.Source: A Derivation of the mismatch response… · p. 24
• The matched template’s amplitude and phase minimize the squared residual norm, yielding the response function in residual form.The displayed least-squares amplitude equality applies only to that amplitude choice.Source: A Derivation of the mismatch response… · p. 24