This repository provides the implementation of Flow-Matching Markov Chain Monte Carlo (FM-MCMC), a novel Bayesian framework designed for Extreme Mass-Ratio Inspiral (EMRI) gravitational wave parameter estimation. The method integrates continuous normalizing flows (CNFs) with parallel tempering MCMC (PT-MCMC), achieving both efficiency and statistical reliability.

The code accompanies our paper:

Unlocking New Paths for Science with Extreme-Mass-Ratio Inspirals: Machine Learning-Enhanced MCMC for Accurate Parameter Inversion
Bo Liang, Chang Liu, Hanlin Song, Zhenwei Lyu, Minghui Du, Peng Xu, et al.

• EMRI signals are prime sources for space-based gravitational-wave detectors (Taiji, LISA).
• Traditional MCMC methods face severe challenges:
  • prohibitively high computational costs,
  • highly multimodal likelihoods with many local maxima,
  • sensitivity to initialization.

FM-MCMC addresses these challenges by combining:

1. Flow Matching (FMPE):

   • Uses continuous normalizing flows to learn high-likelihood regions efficiently.
   • Provides informed initialization to reduce the chance of trapping in local maxima.

2. Parallel Tempering MCMC (PT-MCMC):

   • Performs fine-grained posterior sampling starting from FMPE proposals.
   • Ensures unbiased exploration of the parameter space.

Together, FM-MCMC achieves orders-of-magnitude speed-up while maintaining accurate and unbiased posterior estimation.

• Tested on simulated EMRI signals (Taiji noise model).
• Intrinsic parameters ($M, \mu, a, e_0$) recovered within 1σ posterior intervals for bright EMRI sources ($\text{SNR} > 60$).
• Compared with Eryn (a state-of-the-art PT-MCMC sampler):
  • FM-MCMC converges faster and avoids local maxima,
  • Provides statistically reliable results under realistic noise conditions.

If you use this code, please cite our paper:

@misc{liang2025estimatingorbitalparametersdirect,
      title={Estimating Orbital Parameters of Direct Imaging Exoplanet Using Neural Network}, 
      author={Bo Liang and Hanlin Song and Chang Liu and Tianyu Zhao and Yuxiang Xu and Zihao Xiao and Manjia Liang and Minghui Du and Wei-Liang Qian and Li-e Qiang and Peng Xu and Ziren Luo},
      year={2025},
      eprint={2510.17459},
      archivePrefix={arXiv},
      primaryClass={astro-ph.EP},
      url={https://arxiv.org/abs/2510.17459}, 
}