Factorized neural posterior estimation for rapid and reliable inference of parameterized post-Einsteinian deviation parameters in gravitational waves

Author(s)

Zhang, Yong-Xin, Sun, Tian-Yang, Xiong, Chun-Yu, Liu, Song-Tao, Wang, Yu-Xin, Jin, Shang-Jie, Zhang, Jing-Fei, Zhang, Xin

Abstract

The direct detection of gravitational waves (GWs) by LIGO has strikingly confirmed general relativity (GR), but testing GR via GWs requires estimating parameterized post-Einsteinian (ppE) deviation parameters in waveform models. Traditional Bayesian inference methods like Markov chain Monte Carlo (MCMC) provide reliable estimates but suffer from prohibitive computational costs, failing to meet the real-time demands and surging data volume of future GW detectors. Here, we propose a factorized neural posterior estimation framework: we construct independent normalizing flow models for each of the nine ppE deviation parameters and effectively integrate prior information from other source parameters via a conditional embedding network. Leveraging a hybrid neural network with a convolutional neural network and a Residual Neural Network for feature extraction, our method performs rapid and statistically reliable posterior inference directly from binary black hole signals. Compared to conventional MCMC, our approach achieves millisecond-scale inference time with a speedup factor of $9 \times 10^4$. Comprehensive validations show that the posterior estimates pass the Kolmogorov-Smirnov test and achieve empirical coverage probabilities close to theoretical targets. This work demonstrates the great potential of deep learning for GW parameter estimation and provides a viable technical solution for real-time GR tests with next-generation detectors.

Figures

Caption

Simulated GW signal. The horizontal axis represents time, while the vertical axis represents strain. First, the GW signal is generated. Then, the GW signal within a 1-second time window is extracted from the original signal. Finally, noise processing and whitening are applied to the extracted 1-second GW signal. We have marked the time positions of the peak, window start, and window end with red dashed lines, orange dashed lines, and green dashed lines.

Caption

Model architecture diagram. Both input data and parameters undergo normalization preprocessing. The dashed box on the left represents the hybrid residual embedding network, while the dashed box on the right denotes the conditional embedding network.

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Training strategy principles. The dashed box on the left represents the inner loop for batch data.

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Posterior distributions of GR deviation parameters. The left side of the violin plot shows the results from the deep learning based NPE method, while the right side shows the results from the MCMC method based on the {\tt Dynesty} nested sampler. The black dashed line indicates the true value. The values labeled in the violin plot represent the mean of the posterior distribution obtained by each method. The gray background shows the likelihood function distribution of the parameter. The labels 0PN to 3.5PN on the diagram correspond to $\delta\chi_0$ to $\delta\chi_7$.

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Coverage probability curves of GR deviation parameters from 1000 injections. The labels 0PN to 3.5PN on the diagram correspond to $\delta\chi_0$ to $\delta\chi_7$.

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Coverage probabilities at 68\%, 90\%, and 95\% confidence levels for GR deviation parameters. The horizontal axis represents different GR deviation parameters, while the vertical axis shows coverage probability.