$Φ$asmaNet: A Staged Spectral Token-Refinement Architecture for Fast Gravitational-Wave Signal-to-Noise Ratio Prediction

Ioannis Tsioulis; Anastasios Tefas; Nikolaos Stergioulas

cs.LGNov 3, 2026

$Φ$ asmaNet: A Staged Spectral Token-Refinement Architecture for Fast Gravitational-Wave Signal-to-Noise Ratio Prediction

Authors: Ioannis Tsioulis, Anastasios Tefas, Nikolaos Stergioulas

Organizations: Department of Physics, Aristotle University of Thessaloniki, Greece · School of Computing, Mediterranean College (University of Derby), Athens, Greece · Department of Informatics, Aristotle University of Thessaloniki, Greece

Abstract

Quantifying selection effects in gravitational-wave population studies requires millions of costly signal-to-noise ratio (SNR) calculations. We present $Φ$ asmaNet, a compact neural architecture for structured tabular regression, with staged spectral refinement as an explicit architectural prior. Learned input tokens are refined through ordered attention-based $Φ$ -Stages, each augmented by a $Φ$ -Refractor subspace-mixing module, then revisited through gated top-down $Φ$ -Feedback. Applied to the 11-dimensional parameters of precessing, higher-mode binary black hole signals (IMRPhenomXPHM), it predicts phase-averaged optimal SNR for a single detector at fixed Advanced LIGO design sensitivity. With approximately 732k parameters, it outperforms MLP, 1D-ResNet and Transformer-encoder baselines under matched training across $10^5$ -- $10^6$ samples. At $10^6$ samples, it achieves $100-\mathrm{MAPE}=98.73\%$ on the held-out test set, $99.29\%$ after rank-preserving monotonic (PCHIP) calibration, and $99.39\%$ ( $R^2=0.9998$ ) in the detection-relevant regime $ρ>8$ , with stable accuracy across independent dataset realizations. On a single GPU, it sustains approximately $1.2\times10^5$ SNR evaluations per second and processes a $10^6$ -injection catalogue end-to-end in approximately 8.3 s, about two orders of magnitude faster than conventional waveform-and-inner-product calculations on a 16-core CPU. Exact distance rescaling through $ρ\propto1/d_L$ supports repeated catalogue-wide detectability evaluations. The fully documented pipeline is intended as a reproducible benchmark, while results on public tabular datasets indicate competitiveness beyond gravitational-wave applications. Although demonstrated for a fixed single-detector configuration, the construction supports extensions to detector networks and epoch-specific sensitivities. (Abridged)

Figures & tables

Figure 1: Role of the SNR surrogate. For a set of binary parameters $\theta$ , the optimal SNR $\rho_{\rm opt}$ governs detectability. Evaluating it directly for a large number of injections is computationally demanding and can become a bottleneck in population and selection-effect studies that require very large numbers of evaluations. $\Phi$ asmaNet replaces this per-source computation with a neural-network surrogate that maps $\theta$ directly to $\rho_{\rm opt}$ , enabling high-throughput injection campaigns in which the detectability ( $\rho_{\rm opt}>8$ ), the sensitive volume $V\!T$ , and selection effects are evaluated for large populations.

Figure 2: Four real GWTC events reconstructed with the same waveform model and noise curve used to build the training set. Each curve is the characteristic strain

h_{c}=2f|\tilde{h}(f)|

of the maximum-likelihood posterior sample (IMRPhenomXPHM, projected onto H1 at the event’s sky position and GPS time). The black curve is the aLIGO design noise

h_{n}=\sqrt{fS_{n}(f)}

. The events span the loudest, heaviest, and two unequal-mass systems with and without precession.

Figure 3: Distribution of the training labels: the phase-averaged optimal SNR

\rho_{\rm opt}

at the reference distance

d_{L}=1000

Mpc (shown for the

10^{5}

-sample set), on a linear scale. The label is strongly right-skewed, ranging from

\rho_{\rm opt}\lesssim 0.1

for the lightest, least favorably oriented binaries to

\sim 10^{2}

for the heaviest systems, with a median of

12.3

. This wide dynamic range motivates the

\log_{10}\rho_{\rm opt}

regression target. The single-detector detection threshold

\rho_{\rm opt}=8

is shown as a dashed line. At this reference distance

\approx 64\%

of the samples exceed the detection threshold. The detectability of an astrophysical population follows from the

\rho_{\rm opt}\propto 1/d_{L}

rescaling.

Parameter	Symbol	Distribution	Range
Component masses	$m_{1},m_{2}$	uniform ( $m_{1}\!\geq\!m_{2}$ )	$[3,170]\,M_{\odot}$
(derived)	$\mathcal{M}_{c},\,q$	—	$q\in[1,170/3]$
Spin magnitudes	$\|\bm{s}_{1,2}\|$	uniform	$[0,0.99]$
Spin polar angles	$\theta_{1,2}$	uniform	$[0,\pi]$
Spin azimuthal angles	$\phi_{1,2}$	uniform	$[0,2\pi]$
Inclination	$\cos\iota$	uniform	$[-1,1]$

Table 1: Parameter sampling for the training dataset. Component masses are drawn uniformly and reordered to

m_{1}\geq m_{2}

. The chirp mass

\mathcal{M}_{c}

and mass ratio

q

used as features are derived from each pair. Spins and extrinsic angles are drawn uniformly in the listed variables.

Layer	Operation	Output shape	Notes
Input	–	$[B,11]$	standardized parameter vector $\bm{\theta}$
1	Linear $11\to 256$	$[B,256]$	GELU $\rightarrow$ Dropout ( $p{=}0.1$ )
2	Linear $256\to 256$	$[B,256]$	GELU $\rightarrow$ Dropout ( $p{=}0.1$ )
3	Linear $256\to 256$	$[B,256]$	GELU $\rightarrow$ Dropout ( $p{=}0.1$ )
4	Linear $256\to 256$	$[B,256]$	GELU $\rightarrow$ Dropout ( $p{=}0.1$ )
Output	Linear $256\to 1$	$[B,1]$	predicts $\log_{10}\hat{\rho}$

Table 2: MLPRegressor (width

w{=}256

, depth

d{=}4

) used for SNR regression. Here

B

denotes batch size. Each hidden layer applies Linear

\rightarrow

GELU

\rightarrow

Dropout (

p{=}0.1

Component	Operation	Output shape	Notes
Input	reshape	$[B,1,11]$	11D feature vector $\bm{\theta}$
Stem	Conv $3\times 1$ , $C{=}64$ , stride 1 $\to$ BN $\to$ ReLU	$[B,64,11]$	padding 1
Stage 1	$2\times$ BasicBlock1D , $C{=}64$	$[B,64,11]$	stride 1
Stage 2	$2\times$ BasicBlock1D , $C{=}128$	$[B,128,6]$	stride-2 in first block
Stage 3	$2\times$ BasicBlock1D , $C{=}256$	$[B,256,3]$	stride-2 in first block
Stage 4	$2\times$ BasicBlock1D , $C{=}512$	$[B,512,2]$	stride-2 in first block

Table 3: ResNet1DRegressor architecture (1D ResNet-18-ish, 2 blocks per stage) used for SNR regression. Shapes are shown as

[B,C,L]

until pooling. After pooling we use

[B,D]

. The 11D feature vector is reshaped to a length-11 1D input with one channel. Downsampling is implemented via stride-2 convolutions in the first block of stages 2–4.

Component	Operation	Output shape	Notes
Input	–	$[B,11]$	parameter vector $\bm{\theta}$
Reshape	$\bm{\theta}\mapsto\bm{\theta}[:,:,None]$	$[B,11,1]$	treat each feature as a token
Tokenization	Linear $1\to 128$ (per token)	$[B,11,128]$	scalar $\to$ token embedding
Position enc.	Learnable embeddings	$[B,11,128]$	added to tokens
Encoder	$N_{L}{=}4$ Transformer layers	$[B,11,128]$	pre-LN; MHA: $N_{h}{=}8$ ; FFN: $d_{\mathrm{ff}}{=}256$ ; GELU; dropout $p{=}0.1$
Pooling	Mean over $S$ $\to$ LayerNorm	$[B,128]$	global mean pooling + LN

Table 4: TransformerRegressor architecture used for SNR regression. Shapes are shown as

[B,S,d]

for batch size

B

, sequence length

S{=}11

, and embedding dimension

d=d_{\mathrm{model}}=128

Figure 4:

\Phi

asmaNet overview. A fixed-dimensional structured input is tokenized into a learned token set of shape

(T\times D)

and processed by an ordered sequence of

K

\Phi

-Stages (each

\Phi

-Stage: standard Transformer encoder layer +

\Phi

-Refractor). The model then performs

R

\Phi

-Feedback passes (top-down gated feedback with forward recomputation), before token pooling (readout) and a task-specific head produce the final output.

Component	Operation	Output shape	Notes
Input	–	$[B,11]$	standardized parameter vector $\bm{\theta}$
Tokenizer	Linear $11\to TD$ , reshape	$[B,8,128]$	learned token set $\bm{E}_{0}$
$\Phi$ -Stage ( $\times 3$ )	Enc. layer $+$ $\Phi$ -Refractor	$[B,8,128]$	MHA $N_{h}{=}4$ , FFN $d_{\mathrm{ff}}{=}512$ , GELU; $\Phi$ -Refractor $P{=}4$ ( $D_{p}{=}32$ ), softmax-gated residual; dropout $0.1$
$\Phi$ -Feedback ( $\times 2$ )	Gated update $+$ recompute	$[B,8,128]$	$K{-}1{=}2$ learned gates $+$ parameterless top gate
Readout	Per-token LN $\to$ mean over $T$	$[B,128]$	global mean pooling
Reg. head	Linear $128\to 128\to 1$	$[B,1]$	GELU; outputs $\log_{10}\hat{\rho}$

Table 5:

\Phi

asmaNet architecture used for SNR regression. Shapes are shown as

[B,T,D]

for batch size

B

, token count

T{=}8

, and embedding dimension

D{=}128

Architecture	100k		500k		1M
	Pre	Post	Pre	Post	Pre	Post
$\Phi$ asmaNet	98.32	98.37	98.92	99.16	98.73	99.29
Transformer	$96.39$	$96.79$	$97.04$	$97.43$	$96.03$	$97.40$
ResNet	$97.38$	$97.63$	$97.73$	$97.90$	$97.96$	$98.06$
MLP	$95.62$	$96.14$	$94.67$	$95.98$	$94.49$	$95.64$

Table 6: Accuracy (%) before and after monotonic PCHIP calibration (fit on validation, applied to test), reported side-by-side, for the seed-42 dataset realization and training seed 1337.

\Phi

asmaNet uses the default

\Phi

-Refractor-ON configuration at all sizes. At 100k the OFF variant is marginally better (Table 8 ), so the 100k entry is conservative.

Figure 5: Post-calibration accuracy versus dataset size for all architectures.

Figure 6: Before vs. after calibration performance ( $100-\mathrm{MAPE}$ , %) by model and dataset size. The same monotonic PCHIP calibration procedure is fitted on the validation set and applied uniformly to all models. Differences between pre- and post-calibration bars indicate how much each model benefits from this lightweight correction under the simulated setting considered here.

100k (seed-change check)
Setting	Architecture	Pre	Post
Train seed 2024	$\Phi$ asmaNet (Refractor OFF)	$98.39$	$98.49$
Train seed 2024	Transformer	$96.41$	$96.64$
Train seed 2024	ResNet	$97.58$	$97.70$
Train seed 2024	MLP	$95.73$	$96.34$
1M (dataset-realization check)

Table 7: Robustness checks on test accuracy (

100-\mathrm{MAPE}

, %, pre-/post-calibration): a training-seed change at 100k (all architectures, with

\Phi

asmaNet using

\Phi

-Refractor OFF, the better setting at that scale) and two additional dataset realizations at 1M (generation seeds 43 and 44,

\Phi

asmaNet ).

Configuration	100k		500k		1M
	Pre	Post	Pre	Post	Pre	Post
$\Phi$ -Refractor ON	$98.32$	$98.37$	$98.92$	$99.16$	$98.73$	$99.29$
$\Phi$ -Refractor OFF	$98.56$	$98.57$	$98.73$	$99.22$	$98.61$	$99.23$

Table 8:

\Phi

-Refractor ablation (ON vs. OFF) on the SNR task: test accuracy (

100-\mathrm{MAPE}

, %) before and after monotonic PCHIP calibration, under the otherwise matched protocol of Section 8 . ON is the default

\Phi

asmaNet configuration. All entries use generation seed 42 and training/split seed 1337.

Figure 7: Training and validation loss versus epoch for $\Phi$ asmaNet on the 1M dataset (seed 42, split/training seed 1337), using Huber loss in $\log_{10}\rho$ space with histogram-based weighting by_hist . The trajectories show rapid early improvement followed by a stable plateau under the shared learning-rate scheduling and early-stopping protocol.

Figure 8: Within-tolerance fraction versus epoch for $\Phi$ asmaNet on the 1M dataset (seed 42, split/training seed 1337). We report the fraction of samples with $\left|\rho_{\mathrm{pred}}-\rho_{\mathrm{true}}\right|/\rho_{\mathrm{true}}\leq 1\%$ in linear-SNR space on train and validation splits. This strict metric stabilizes at later epochs, consistent with convergence under the matched training protocol.

Panel of Figure 9 — Figure 9: Test residual histograms for $\Phi$ asmaNet on the 1M dataset (seed 42, split/training seed 1337) in linear-SNR space. Left: raw residuals ( $\rho_{\mathrm{pred}}-\rho_{\mathrm{true}}$ ). Right: residuals after monotonic PCHIP calibration (fit on validation, applied to test).

Figure 10: Calibrated test residuals versus true SNR for $\Phi$ asmaNet on the 1M dataset (seed 42, split/training seed 1337), shown in linear-SNR space. This plot checks for SNR-dependent structure after applying the monotonic PCHIP mapping (fit on validation, applied to test).

Figure 11: Predicted versus true SNR for $\Phi$ asmaNet on the 1M test set (seed 42, split/training seed 1337), shown in linear-SNR space (raw predictions).

Figure 12: Predicted versus true SNR for $\Phi$ asmaNet on the 1M test set (seed 42, split/training seed 1337), shown in linear-SNR space after monotonic PCHIP calibration (fit on validation, applied to test).

Architecture	Params	Relative	Key sizing
MLP	$\sim$ 201 k	$1.0\times$	width $w{=}256$ , depth $d{=}4$
Transformer	$\sim$ 540 k	$2.7\times$	$d_{\mathrm{model}}{=}128$ , $N_{L}{=}4$ , $N_{h}{=}8$ , $d_{\mathrm{ff}}{=}256$
$\Phi$ asmaNet	$\sim$ 732 k	$3.7\times$	$T{=}8$ , $D{=}128$ , $K{=}3$ , $R{=}2$ , $P{=}4$
ResNet (1D)	$\sim$ 3.98 M	$19.8\times$	width $w{=}64$ , 4 stages $\times$ 2 blocks

Table 9: Trainable parameter counts for each architecture under the default configurations used in this work. All counts refer to the models as instantiated for the 11-feature SNR regression task.

$\Phi$ asmaNet variant	Accuracy	Cross-entropy
$\Phi$ -Refractor ON	$0.9664\pm 0.0002$	$0.2828\pm 0.0049$
$\Phi$ -Refractor OFF	$0.9634$	–

Table 10: Covertype test performance of

\Phi

asmaNet (mean

\pm

SEM over seeds). No hyperparameter tuning was performed. The configuration is inherited from the SNR regression experiments in Section 6 .

$\Phi$ asmaNet variant	LogLoss $\times 100$	Accuracy	ROC-AUC
Default (Refractor ON )	$25.9741\pm 0.1289$	$0.8915\pm 0.0005$	$0.9950\pm 0.0003$

Table 11: Epsilon test performance of

\Phi

asmaNet (mean

\pm

SEM over seeds). Log loss is scaled by

\times 100

as is customary for this dataset. No hyperparameter tuning was performed. The configuration is inherited from the SNR regression experiments in Section 6 .

Gravitational-Wave Astronomy

Gravitational-wave astronomy uses ripples in spacetime to detect cosmic events, measure compact objects, and test gravity.

See all 21 papers

AI-enabled gravitational-waves searches for binary neutron stars at optimal sensitivity
Jul 1, 2026Bhavya Gupta, Deep Chatterjee, William Benoit +7Gravitational-Wave Astronomy
1LIGO Laboratory, 185 Albany St, MIT, Cambridge, MA 02139, USA · Department of Physics, MIT, Cambridge, MA 02139, USA · School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
Survey of Novel Deep Learning Architectures for Denoising Gravitational-wave Signals
Sep 7, 2026Rohan Raha, Prayush KumarGravitational-Wave Astronomy
Department of Physics, Indian Institute of Science, Bangalore 560012, Karnataka, India · International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
Testing General Relativity Through Gravitational Wave Classification: A Convolutional Neural Network Framework
May 4, 2026Lavinia Heisenberg, Shayan Hemmatyar, Hector Villarrubia-RojoGravitational-Wave Astronomy Binary Classification
Institute for Theoretical Physics, Heidelberg University, Philosophenweg 16, 69120 Heidelberg, Germany · Departamento de F´ısica Te´orica and IPARCOS, Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain

Jul 1, 2026J/K move · Enter open · S save

AI-enabled gravitational-waves searches for binary neutron stars at optimal sensitivity

Bhavya Gupta, Deep Chatterjee, William Benoit, Ethan Marx+6

1LIGO Laboratory, 185 Albany St, MIT, Cambridge, MA 02139, USA · Department of Physics, MIT, Cambridge, MA 02139, USA · School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

Gravitational-Wave Astronomy

Open paper AskPDF arXiv

Gravitational Waves (GWs) represent the newest window of astronomy, furthering our understanding of compact objects like black holes and neutron stars in the Universe. The signal from two merging neutron stars is especially interesting since it brings the prospect of concordant electromagnetic and neutrino emissions. Such multi-messenger observations have a transformational impact on fundamental physics, nuclear matter, astrophysics, and gravity. It was first witnessed in 2017 with the detection of the binary neutron star (BNS) merger GW170817. However, searching for BNS signals in real-time in the LIGO-Virgo-KAGRA (LVK) GW detectors presents a computational challenge, as the data streaming out must be matched against

\sim

million reference waveforms, which requires up to a thousand CPU cores. We present a different approach using neural networks to learn the presence of a signal in the data. Our algorithm, called Aframe, was deployed in the LVK's fourth observing run and was the first artificial intelligence (AI)-enabled search to detect multiple binary black holes (BBHs) live. In this work, we demonstrate that the approach extends to the lower-mass BNS regime, and is the first AI-enabled search that achieves sensitivity comparable to matched-filter pipelines at lower computational and latency costs. The challenge of the longer-duration BNS signals is addressed by heterodyning the data, following which the network architecture used for BBHs is sufficient to distinguish signal versus background. We also show that this analysis requires a single non-flagship GPU for online deployment. Furthermore, the design and adoption of inference-as-a-service tools allow rapid offline analysis using a distributed pool of GPU resources. Hence, aside from the use case of rapid online data analysis, we also establish the use of Aframe for efficient archival data analysis.

Abstract

Gravitational-Wave Astronomy

Explore similar work

AI-enabled gravitational-waves searches for binary neutron stars at optimal sensitivity

Survey of Novel Deep Learning Architectures for Denoising Gravitational-wave Signals

Testing General Relativity Through Gravitational Wave Classification: A Convolutional Neural Network Framework