2024年5月17日, 17:05-17:20 

2024 年重庆引力与天体物理学术研讨会 · 重庆邮电大学

Exploring the Frontiers of Parameter Estimation with AI in Gravitational Wave Research

王赫 (He Wang)

hewang@ucas.ac.cn

中国科学院大学 · 国际理论物理中心（亚太地区）

中国科学院大学 · 引力波宇宙太极实验室（北京/杭州）

In cooperation with

Z.Cao, Z.Ren, M.Du, B.Liang, P.Xu, Z.Luo, Y.Wu, et al.

  • non-GW
    • NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport (1903.03704 )
    • Accelerated Bayesian inference using deep learning (https://doi.org/10.1093/mnras/staa1469)
    • Nested Sampling Methods (2101.09675)
    • pocoMC: A Python package for accelerated Bayesian inference in astronomy and cosmology (2207.05660)
    • Parallelized Acquisition for Active Learning using Monte Carlo Sampling (2305.19267)
    • NAUTILUS: boosting Bayesian importance nested sampling with deep learning (2306.16923)
    • Improving Gradient-guided Nested Sampling for Posterior Inference (2312.03911)
    • floZ: Evidence estimation from posterior samples with normalizing flows (2404.12294)
    • Deep Learning and genetic algorithms for cosmological Bayesian inference speed-up (2405.03293)
  • GW
    • Nested Sampling with Normalising Flows for Gravitational-Wave Inference (2102.11056)
    • Bilby-MCMC: an MCMC sampler for gravitational-wave (2106.08730)
    • Nested sampling for physical scientists (2205.15570)
    • Fast gravitational wave parameter estimation without compromises (2302.05333)
    • Importance nested sampling with normalising flows (2302.08526)
    • Neural density estimation for Galactic Binaries in LISA data analysis (2402.13701)
    • Robust parameter estimation within minutes on gravitational wave signals from binary neutron star inspirals (2404.11397)
  •  

10+5 = 15min

space-based (2)

what is flow and flow-based (4)

how flow can be used in MCMC.

mini Global-fit + flow

Taiji

Tianqin

https://twitter.com/chipro/status/1768388213008445837?s=46&t=JmDXWgIucgr_FlsBFTvuRQ

DINGO+SEOBNRv4EHM找了3个ebbh

Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA
https://dcc.ligo.org/LIGO-G2400750

BEFORE

AFTER

LIGO-G2300554

  • In 1916, A. Einstein proposed the GR and predicted the existence of GW.

  • Gravitational waves (GW) are a strong field effect in the GR.

    • 2015: the first experimental detection of GW from the merger of two black holes was achieved.

    • 2017: the first multi-messenger detection of a BNS signal was achieved, marking the beginning of multi-messenger astronomy.

    • 2017: the Nobel Prize in Physics was awarded for the detection of GW.

    • As of now: more than 90 gravitational wave events have been discovered.

    • O4, which began on May 24th 2023, is currently in progress.

Gravitational Wave Astronomy

Gravitational waves generated by binary black holes system

GW detector

Gravitational Wave Astronomy

  • Fundamental Physics
    • Existence of gravitational waves
    • To put constraints on the properties of gravitons
  • Astrophysics
    • Refine our understanding of stellar evolution
    • and the behavior of matter under extreme conditions.
  • Cosmology
    • The measurement of the Hubble constant
    • Dark energy

The first GW event of GW150914

Parameter estimation · Scientific discovery

Credit: LIGO Magazine.

  • Traditional parameter estimation (PE) techniques rely on Bayesian analysis methods (posteriors + evidence)

  • Computing the full 15-dimensional posterior distribution estimate is very time-consuming:
    • Calculating likelihood function
    • Template generation time-consuming
  • Machine learning algorithms are expected to speed up! 

Challenges of Parameter Estimation for GW

Bayesian statistics

Data quality improvement

Credit: Marco Cavaglià 

LIGO-Virgo data processing

GW searches

Astrophsical interpretation of GW sources

CQG. 37 (2020) 055002

AI for Gravitational Wave: Parameter Estimation

  • A complete 15-dimensional posterior probability distribution, taking about 1 s (<< \(10^4\) s).
  • Prior Sampling: 50,000 Posterior samples in approximately 8 Seconds.
  • Capable of calculating evidence
  • Processing time: (using 64 CPU cores)
    • less than 1 hour with IMRPhenomXPHM,
    • approximately 10 hours with SEOBNRv4PHM

PRL 127, 24 (2021) 241103.

PRL 130, 17 (2023) 171403.

Nature Physics 18, 1 (2022) 112–17

HW, et al. Big Data Mining and Analytics 5, 1 (2021) 53–63.

A diagram of prior sampling between feature space and physical parameter space

Rapid PE for Space-borne GW Detection

  • Data curation

    • Model: frequency domain, PhenomD,  TDI-A,E response

    • Data:1 day, 15s per sample, shape=(2, 3, 2877)

    • Noise: Gaussian stationary from PSD + GB confusion noise

    • Project: Taiji program

M. Du, B. Liang, HW, P. Xu, Z. Luo, Y. Wu. SCPMA 67, 230412 (2024).

  • Motivation: To preprocess Global Fit data for early detection of merged electromagnetic observations for MBHBs.

（Based on 1912.02762）

The ABC of Normalizing Flow 

The main idea of flow-based modeling is to express \(\mathbf{y}\in\mathbb{R}^D\) as a transformation \(T\) of a real vector \(\mathbf{z}\in\mathbb{R}^D\) sampled from \(p_{\mathrm{z}}(\mathbf{z})\):

\mathbf{y}=T(\mathbf{z}) \quad \text { where } \quad \mathbf{z} \sim p_{\mathrm{y}}(\mathbf{z})

Note: The invertible and differentiable transformation \(T\) and the base distribution \(p_{\mathrm{z}}(\mathbf{z})\) can have parameters \(\{\boldsymbol{\phi}, \boldsymbol{\psi}\}\) of their own, i.e. \( T_{\phi} \) and \(p_{\mathrm{z},\boldsymbol{\psi}}(\mathbf{z})\).

Change of Variables:

p_{\mathrm{y}}(\mathbf{y})=p_{\mathrm{z}}(\mathbf{z})\left|\operatorname{det} J_{T}(\mathbf{z})\right|^{-1} \quad \text { where } \quad \mathbf{u}=T^{-1}(\mathbf{x}) .
J_{T}(\mathbf{z})=\left[\begin{array}{ccc} \frac{\partial T_{1}}{\partial \mathrm{z}_{1}} & \cdots & \frac{\partial T_{1}}{\partial \mathrm{z}_{D}} \\ \vdots & \ddots & \vdots \\ \frac{\partial T_{D}}{\partial \mathrm{z}_{1}} & \cdots & \frac{\partial T_{D}}{\partial \mathrm{z}_{D}} \end{array}\right]

Equivalently,

The Jacobia \(J_{T}(\mathbf{u})\) is the \(D \times D\) matrix of all partial derivatives of \(T\) given by:

p_{\mathrm{y}}(\mathbf{y})=p_{\mathrm{z}}\left(T^{-1}(\mathbf{y})\right)\left|\operatorname{det} J_{T^{-1}}(\mathbf{y})\right|
p_{\mathrm{y}}(\mathbf{y})
p_{\mathrm{z}}(\mathbf{z})
\mathbf{z}
\mathbf{y}
T
T^{-1}

base density

target density

Rational Quadratic Neural Spline Flows (RQ-NSF)

（Based on 1912.02762）

  • Data: target data \(\mathbf{y}\in\mathbb{R}^{11}\) (with condition data \(\mathbf{x}\)).
  • Task:
    • Fitting a flow-based model \(p_{\mathrm{y}}(\mathbf{y} ; \boldsymbol{\theta})\) to a target distribution \(p_{\mathrm{y}}^{*}(\mathbf{y})\)
    • by minimizing KL divergence with respect to the model’s parameters \(\boldsymbol{\theta}=\{\boldsymbol{\phi}, \boldsymbol{\psi}\}\),
    • where \(\boldsymbol{\phi}\) are the parameters of \(T\) and \(\boldsymbol{\psi}\) are the parameters of \(p_{\mathrm{z}}(\mathbf{z})=\mathcal{N}(0,\mathbb{I})\).
  • Loss function:




     
  • Assuming we have a set of samples \(\left\{\mathbf{y}_{n}\right\}_{n=1}^{N}\sim p_{\mathrm{y}}^{*}(\mathbf{y})\),



    Minimizing the above Monte Carlo approximation of the KL divergence is equivalent to fitting the flow-based model to the samples \(\left\{\mathbf{y}_{n}\right\}_{n=1}^{N}\) by maximum likelihood estimation.
\mathcal{L}(\boldsymbol{\theta}) \approx-\frac{1}{N} \sum_{n=1}^{N} \log p_{\mathrm{z}}\left(T^{-1}\left(\mathbf{y}_{n} ; \boldsymbol{\phi}\right) ; \boldsymbol{\psi}\right)+\log \left|\operatorname{det} J_{T^{-1}}\left(\mathbf{y}_{n} ; \boldsymbol{\phi}\right)\right|+\mathrm{const.}
\begin{aligned} \mathcal{L}(\boldsymbol{\theta}) &=D_{\mathrm{KL}}\left[p_{\mathrm{y}}^{*}(\mathbf{y}) \| p_{\mathrm{y}}(\mathbf{y} ; \boldsymbol{\theta})\right] \\ &=-\mathbb{E}_{p_{\mathbf{y}}^{*}(\mathbf{y})}\left[\log p_{\mathbf{y}}(\mathbf{y} ; \boldsymbol{\theta})\right]+\text { const. } \\ &=-\mathbb{E}_{p_{\mathbf{y}}^{*}(\mathbf{y})}\left[\log p_{\mathrm{z}}\left(T^{-1}(\mathbf{y} ; \boldsymbol{\phi}) ; \boldsymbol{\psi}\right)+\log \left|\operatorname{det} J_{T^{-1}}(\mathbf{y} ; \boldsymbol{\phi})\right|\right]+\mathrm{const} . \end{aligned}
\mathbb{E}_{p_{\mathbf{y}}^{*}(\mathbf{y})}\left[\log p_{\mathbf{y}}^{*}(\mathbf{y} ; \boldsymbol{\theta})\right]
\vec\theta = (m_1,m_2,d_L, ...) \in P_{prior}
\vec{x}=\vec{h}_{\vec{\theta}} + \vec{n}

nflow

\vec{z} \Rightarrow \mathcal{N}(0,\mathbb{I})
\vec\theta = (m_1,m_2,d_L, ...) \in P_{posterior}
\vec{x}=\vec{h}_{\vec{\theta}} + \vec{n}

nflow

\vec{z} \in \mathcal{N}(0,\mathbb{I})

Train

Test

The ABC of Normalizing Flow 

Rapid PE for Space-borne GW Detection

  • Results:

M. Du, B. Liang, HW, P. Xu, Z. Luo, Y. Wu. SCPMA 67, 230412 (2024).

  • Computational performance

    • 10000 samples in 2.7 sec

  • Multimodality in extrinsic parameters

  • Unbiased estimation and confidence validation

Ongoing and Future Projects

Pipeline Targets Programing Language (sampling method) Comments
GLASS 
(Littenberg&Cornish 2023)		 Noise,
UCB, VGB, MBHB		 C / Python (TPMCMC / RJMCMC) noise_mcmc+gb_mcmc+vb_mcmc+global_fit
Eryn UCB Python (TPMCMC / RJMCMC) Mini code for UCB case
PyCBC-INFERENCE MBHB Python (?) Unavailable
Bilby in Space / tBilby MBHB / ? ? / Python? (RJMCMC) Unavailable
Strub et al. UCB ? (GP) Unavailable / GPU-based
Zhang et al. (LZU) UCB ? (PSO) MLP
Balrog MBHB ?

(Sec.8.6 Red Book)

Global Fit

  • The idea of the global fit method is to comprehensively model all astrophysical and instrumental features present in the space-borne gravitational wave data.
  • This approach not only focuses on the signal from a single source, but attempts to capture the combined effects of all sources in the data, conducting a comprehensive analysis of the entire dataset to identify and model all potential signal and noise sources.

Technical challenges:

  • High dimensional
  • Highly correlated
  • Multimodality
  • Trans-dimensional

Text

Ongoing and Future Projects

Neural density estimation

  • Density fit for posterior distributions
    • use the old posterior to form a proposal for the extended data.
  • Density fit for the Galaxy
    • fitt a Galaxy model for joint distribution for \((A, \beta, \lambda)\).
  • ...

Text

Ref:

  • Ashton, G, and C Talbot. MNRAS 507, no. 2 (2021): 2037–51.
  • Korsakova, N, et al. (2402.13701)
  • Wouters, T, et al. (2404.11397​)

Ongoing and Future Projects

Neural density estimation

  • Density fit for posterior distributions
    • use the old posterior to form a proposal for the extended data.
  • Density fit for the Galaxy
    • fitt a Galaxy model for joint distribution for \((A, \beta, \lambda)\).
  • ...

Text

nflow

\mathcal{N}(0,\mathbb{I})

Ref:

  • Ashton, G, and C Talbot. MNRAS 507, no. 2 (2021): 2037–51.
  • Korsakova, N, et al. (2402.13701)
  • Wouters, T, et al. (2404.11397​)

Ongoing and Future Projects

Neural density estimation

  • Density fit for posterior distributions
    • use the old posterior to form a proposal for the extended data.
  • Density fit for the Galaxy
    • fitt a Galaxy model for joint distribution for \((A, \beta, \lambda)\).
  • ...

Text

nflow

\mathcal{N}(0,\mathbb{I})
for _ in range(num_of_audiences):
    print('Thank you for your attention! 🙏')

This slide: https://slides.com/iphysresearch/2024may_cqupt