Identifying lensed gravitational waves with physics-informed posterior learning

Technical Summary: Identifying Lensed Gravitational Waves with Physics-Informed Posterior Learning

Problem Statement
The detection of gravitational waves (GWs) from compact-binary coalescences offers a pathway to test gravity and cosmology, but identifying gravitationally lensed events remains a significant challenge. While lensed events could provide repeated views of the same merger to constrain dark matter structures and cosmological distances, distinguishing them from unrelated multiple-merger events occurring within the same analysis window is difficult. Current methods struggle when a lensed multi-image signal is confused with unrelated mergers that happen to overlap in time. Existing approaches either rely on posterior overlaps and time-delay consistency (which can be computationally expensive or ambiguous) or single-event signatures (like microlensing). Deep learning classifiers offer speed but often lack explicit control over the physical source-consistency that distinguishes lensed images from unrelated events. The goal is to construct an event ranking system that effectively separates lensed multi-image signals from physically relevant unrelated multiple-merger events without requiring prior knowledge of the lens potential, image count, or time delays.

Methodology
The authors propose a "physics-informed posterior learning" framework that leverages geometric-optics consistency. In this regime, lensing alters image amplitudes, arrival times, and Morse phase offsets but preserves the intrinsic phase evolution of the source. The method focuses on a low-dimensional projection of the source parameters: the detector-frame chirp mass ($M_{c,z}$) and the symmetric mass ratio ($\eta$).

The architecture consists of three integrated components:

1. Direct Waveform Classifier: A one-dimensional residual network (ResNet) that maps whitened time-series data from the LIGO Hanford (H1) and Livingston (L1) detectors directly to a lensing score. This serves as a baseline measuring performance based solely on waveform morphology.
2. Neural Posterior Estimator (NPE): A simulation-trained model that infers an approximate posterior distribution, $q_\phi(\vartheta_{ph}|x)$, over the common-source parameters ($\log M_{c,z}, \eta$) for a given waveform. This estimator is trained exclusively on lensed multi-image simulations. At inference, it generates posterior samples for both lensed and unrelated events. For unrelated events, the "forced" common-source hypothesis results in a broad or split posterior, whereas lensed events yield a compact posterior consistent with a single source.
3. Fusion Classifier: An attention-gated network that combines the direct waveform features with the posterior samples. The posterior samples are treated as an unordered set, embedded via a point-wise MLP, and aggregated using permutation-invariant pooling. These features are then fused with the direct waveform features via cross-attention and a learned sigmoid gate. This allows the model to utilize source-consistency information to re-rank ambiguous events while retaining waveform evidence when posterior inference is noisy.

Training and Evaluation Strategy
The system is trained using generic multi-image simulations where the number of images (2–5), amplitude factors, time delays, and Morse phases are sampled from broad priors, without labeling specific physical lens families. This ensures the model learns general source-consistency rather than specific lens morphologies.

• Held-out Evaluation: The model is tested on four specific physical lens families—Point Mass (PM), Singular Isothermal Sphere (SIS), Singular Isothermal Ellipsoid (SIE), and SIE with external shear (SIE+Shear)—which were not used as training labels. This tests the generalization of the ranking task across different lens potentials.
• Data: Simulations inject non-spinning binary black hole (BBH) signals into pre-vetoed O4 background segments from the GW Open Science Center.
• Metrics: Performance is evaluated using ROC-AUC, PR-AUC, and detection efficiency (recall) at fixed false-positive rates (FPR), specifically 1% and 0.1%.

Key Results

• Performance Gains: The attention-gated fusion classifier outperforms the direct waveform baseline. Averaged over the four held-out lens families, the fusion model achieves a ROC-AUC of 0.844 compared to 0.785 for the direct classifier.
• Detection Efficiency: At a 1% reference FPR threshold, the fusion method raises detection efficiency from 20.8% to 35.2% for the observationally motivated BBH population.
• SNR Reduction: The network signal-to-noise ratio (SNR) required for 50% detection efficiency is lowered from 45.3 to 33.5. This corresponds to a 1.35 times increase in the SNR-equivalent distance scale ($D_{50}$).
• Generalization: The gains are most pronounced for SIE and SIE+Shear lenses, where ellipticity and shear broaden image configurations, making source consistency less tied to specific image geometries. The posterior-set branch alone performs poorly (ROC-AUC ~0.59), confirming that its value lies in the fusion with waveform morphology.
• Limitations: The improvement is limited by "loud" unrelated multiple-merger events that are partly source-consistent (i.e., their component masses happen to align in the mass plane). The false-positive tail is dominated by these specific, temporally compact, and high-SNR unrelated events.

Significance and Claims
The paper claims that physical consistency can serve as a guiding principle for machine learning searches in dense gravitational-wave catalogs. By encoding the geometric-optics fact that lensing preserves intrinsic phase evolution, the method provides a robust ranking statistic that generalizes across different lens families without requiring specific lens model labels during training.

The authors emphasize that this is a controlled study using non-spinning BBHs and H1/L1 data. They caution that the reported false-alarm rates are diagnostics based on specific simulated populations and must be recalibrated for real-world searches using the actual unrelated multiple-merger population of the target catalog. The study suggests that future improvements could come from incorporating sky-position consistency (detector-response consistency) as network localization improves with more detectors (e.g., KAGRA, LIGO-India), which would further constrain the "unrelated multiple-merger" hypothesis. Ultimately, the work demonstrates that simulation-trained approximate posteriors can effectively bridge the gap between fast waveform classification and rigorous physical consistency checks.