Detection of Lensed Gravitational Waves in the Millihertz Band Using Frequency-Domain Lensing Feature Extraction Network

TL;DR

This work tackles the challenge of rapidly identifying lensed gravitational waves in the millihertz band, where wave-optics effects are significant and conventional Bayesian searches are computationally expensive. It introduces the Dual-Channel Lensing feature extraction eXtended Long Short-Term Memory Network (DCL-xLSTM), which processes frequency-domain amplitude data from the LISA A and E TDI channels to capture diffraction patterns across the wave-optics to geometric-optics transition, using two memory types (matrix-valued and vector-valued) with memory mixing. The method is validated on simulations with Point Mass and Singular Isothermal Sphere lenses, showing an AUC of about 0.991 and TPR above 98% at FPR below 1%, outperforming standard RNN/LSTM baselines and maintaining robustness across lens masses and SNR. These results suggest a scalable, high-efficiency approach for screening large future GW datasets for lensed events, enabling faster cosmological and fundamental-physics investigations with space-based detectors like LISA, Taiji, and TianQin.

Abstract

The space-based gravitational wave (GW) detectors are expected to observe lensed GW events, offering new opportunities for cosmology and fundamental physics. In the millihertz frequency band, the GW wavelength is often comparable to the Schwarzschild radius of the lens, where wave-optics effects are significant. Although traditional matched filtering is effective, the intense computational resources required motivate the search for more efficient alternatives to accelerate candidate event screening. To address this bottleneck, we introduce a Dual-Channel Lensing feature extraction eXtended Long Short-Term Memory Network (DCL-xLSTM). Unlike conventional recurrent architectures, DCL-xLSTM employs a matrix-valued memory structure and a memory-mixing mechanism to effectively capture amplitude diffraction patterns that span the entire millihertz frequency band. Trained on data generated by Point Mass (PM) and Singular Isothermal Sphere (SIS) models accounting for the transition from wave-optics to geometric-optics, the proposed method achieves an area under the curve (AUC) exceeding 0.99, maintaining a true positive rate (TPR) above 98% at a false positive rate (FPR) below 1%. The network is robust against variations in signal-to-noise ratio, lens type, and lens mass, establishing its viability as a high-efficiency tool for future space-based GW detection.

Figures (8)

• Figure 1: Overview of gravitational lensing regimes and signal amplification.Left: The lens mass ($M_{L}$) versus GW frequency ($f$) parameter space. The dashed line ($w=1$) marks the transition between geometric and wave optics, with the shaded orange band ($0.1 < w < 10$) . Sensitivity bands for LISA (blue) and LIGO (gray) are shown for references. Right: The frequency amplification factor $|F(w)|$ is a function of dimensionless frequency $w$ for Point Mass (PM, solid) and Singular Isothermal Sphere (SIS, dashed) models with a source impact parameter $y=0.3$. The gray shaded region corresponds to the transition regime shown in the left panel.
• Figure 2: A schematic diagram of gravitational lensing of GWs. The signal from binary system is deflected by an intervening lens. Distances are shown: source-to-lens ($D_{LS}$), lens-to-observer ($D_L$). The impact parameter of the source relative to the lens axis is $\gamma$, and $\xi_0$ is the Einstein radius in the lens plane.
• Figure 3: The architecture of the DCL-xLSTM for GW classification. Frequency-domain strain amplitudes from the A and E TDI channels are preprocessed and sampled at 2048 points to form a dual-channel input sequence $\{\mathbf{x}_t\}$, where $\mathbf{x}_t = (|A(f_t)|, |E(f_t)|)$. The sequence is processed by a stack of mLSTM and sLSTM blocks, which extract long-range spectral features and cross-channel correlations characteristic of lensing. The final hidden representation is passed through a fully connected layer to produce a probability for lensed versus unlensed gravitational-wave events.
• Figure 4: Receiver operating characteristic (ROC) curves for the binary classification task on the combined dataset. The DCL-xLSTM model (red solid line, AUC = 0.991) demonstrates performance, significantly outperforming the LSTM (blue dashed line, AUC = 0.920) and the RNN (green dashed line, AUC = 0.785). The gray dashed line represents the random classifier baseline (AUC = 0.5). The x-axis (False Positive Rate) is plotted on a logarithmic scale to highlight performance at low FPRs, which is critical for rare event detection.
• Figure 5: Comparative ROC curves for GW signals classification. The plot illustrates the performance of DCL-xLSTM (red), LSTM (blue), and RNN (green) models against the Higher Mass (solid lines) and Lower Mass (dashed lines) datasets. While all models exhibit improved sensitivity for higher lens masses (solid curves), the DCL-xLSTM model displays stability, showing minimal performance degradation between mass regimes (AUC decreases only from 0.997 to 0.993). In contrast, the LSTM and RNN models show more significant performance gaps between the two datasets, highlighting the superior generalization capability of the DCL-xLSTM architecture.