Aug 2, NZ Gravity seminar @UoA

Enhancing Gravitational Wave Astronomy with Artificial Intelligence

He Wang (王赫)

hewang@ucas.ac.cn

International Centre for Theoretical Physics Asia-Pacific (ICTP-AP), UCAS

Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou), UCAS

On behalf of the LIGO-VIRGO-KAGRA collaborations

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

  • 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 waves generated by binary black holes system

GW detector

LIGO-VIRGO-KAGRA network

2017 Nobel Prize in Physics

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

  • Detecting gravitational waves require a mix of FIVE key ingredients:
    1. good detector technology
    2. good waveform predictions
    3. good data analysis methodology and technology
    4. coincident observations in several independent detectors
    5. coincident observations in electromagnetic astronomy

—— Bernard F. Schutz

​​DOI: 10.1063/1.1629411 

GWTC-3

Gravitational Wave Astronomy

Technical Challenges: Data Processing for GW

GW Data characteristics







 

 

 

  • ​Noise: non-Gaussian and non-stationary

  • Signal:

    • (Earth-based) A low signal-to-noise ratio (SNR) which is typically about 1/100 of the noise amplitude (-60 dB).

    • (Space-based) A superposition of all GW signals (e.g.: \(10^4\) of GBs, \(10\sim10^2\) of SMBHs, and \(10\sim10^3\) of EMRIs, etc.) received during the mission's observational run.

Matched filtering techniques
 

  • In Gaussian and stationary noise environments, the optimal linear algorithm for extracting weak signals

  • Works by correlating a known signal model \(h(t)\) (template) with the data.
  • Starting with data: \(d(t) = h(t) + n(t)\).
  • Defining the matched-filtering SNR \(\rho(t)\):
    \(\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2 \) , where \(\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df \) ,
    \(\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df \), \(S_n(f)\) is noise power spectral density (one-sided).

Text

LIGO-VIRGO-KAGRA

LISA / Taiji project

Text

  • Frequentist hypothesis testing and likelihood princple:

    • make some assumptions about signal and noise hypothesis

    • write down the likelihood function for a signal in noise

    • find the parameters that maximise it

    • define a corresponding detection statistic

      \(\rightarrow\) recover the MF
  • Bayesian hypothesis testing:
    • start from the same likelihood
    • define a prior over signal parameters
    • marginalise over them to arrive at a Bayes factor
    • Often the dirty secret: just treat this as a Frequentist detection statistic
      \(\rightarrow\) recover the MF (for certain prior choices)

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

Text

Pioneering works utilizing CNN

  • The most common and direct approach, from Computer Vision (CV) to GW signal processing: pixel point \(\Rightarrow\) sampling point.

 

 





 

  • Convolutional neural networks (CNN) can achieve comparable performance to Matched Filtering and surpass them in terms of execution speed (with GPU support) under Gaussian stationary noise.

AI for Science \(\rightarrow\) AI for GW Astronomy






 

 

 

 




 

  • Artificial Intelligence (AI) has great potential to revolutionize gravitational wave astronomy by improving data analysis, modeling, and detector development.
  • Representation and supervised learning crucially extract features from GW signals, autonomously identifying informative features and leveraging labeled data for accuracy.

Text

Exported: Oct, 2023 (in preparation)

PRL, 2018, 120(14): 141103.

PRD, 2018, 97(4): 044039.

GW Data Analysis with Artificial Intelligence

Matched-filtering Convolutional Neural Network (MFCNN)

Text

  • GW templates can be utilized as recognizable features for signal detection.
  • It is feasible to generalize both matched-filtering and neural networks.
  • Linear filters (i.e., matched-filtering) in signal processing can be reformulated as neural layers (i.e., CNNs).

MLGWSC-1

  • The majority of AI algorithms used for testing are highly sensitive to non-Gaussian real noise backgrounds, resulting in high false positive rates.

(MFCNN group) H.W., et al. PRD (2023)

Text

CL.M., W.W., H.W., et al. PRD (2022)

Ensemble learning

  • Leverages statistical approaches to utilize more information for making informed decisions by combining multiple models.

Real-time GW searches for GW150914

H.W., et al. PRD (2020)

Text

Expanding the dimension of the output

  • is to call more information to make decisions in improving AI models.

Text

CL.M., W.W., H.W., et al. PRD (2023)

GW Data Analysis with Artificial Intelligence

GW search · Pipeline

Text

Beyond Speed: Generalization and Discovery in GW Detection

  • Our primary goal is not speed but the model's ability to generalize and discover new GW signals, including those beyond the reach of matched filtering techniques and General Relativity (GR).
  • Leveraging our experience in  signal modeling  (MFCNN) and noise modeling (WaveFormer), we are gradually building an offline pipeline capable of searching for signals in complete GW observation data and calculating FARs.

Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012)

Feature extraction

Convolutional Neural Network (ConvNet or CNN)

Classification

GW150914

GW151226

GW151012

  • CNNs always works pretty good on stimulated noises.
  • However, when on real noises from LIGO, this approach does not work that well.
  • It's too sensitive against the background + hard to find GW events)
>> Is it matched-filtering ?
>> Wait, It can be matched-filtering!

GW search · Pipeline

Text

Beyond Speed: Generalization and Discovery in GW Detection

  • Our primary goal is not speed but the model's ability to generalize and discover new GW signals, including those beyond the reach of matched filtering techniques and General Relativity (GR).
  • Leveraging our experience in  signal modeling  (MFCNN) and noise modeling (WaveFormer), we are gradually building an offline pipeline capable of searching for signals in complete GW observation data and calculating FARs.

He Wang, et al. PRD 101, 10 (2020): 104003

GW150914

GW151226

GW151012

 MFCNN
 MFCNN
 MFCNN

Matched-filtering (cross-correlation with the templates) can be regarded as a convolutional layer with a set of predefined kernels.

  • In practice, we use matched filters as an essential component of feature extraction in the first part of the CNN for GW detection.

Real-time GW searches for GW150914

GW search · Pipeline

Text

Beyond Speed: Generalization and Discovery in GW Detection

  • Our primary goal is not speed but the model's ability to generalize and discover new GW signals, including those beyond the reach of matched filtering techniques and General Relativity (GR).
  • Leveraging our experience in  signal modeling  (MFCNN) and noise modeling (WaveFormer), we are gradually building an offline pipeline capable of searching for signals in complete GW observation data and calculating FARs.

He Wang, et al. PRD 101, 10 (2020): 104003

  • Transform matched-filtering method from frequency domain to time domain.
  • The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

Frequency domain

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)

Time domain

(normalizing)

(matched-filtering)

\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)

\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)

(whitening)

where

\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix} \bar{d}(t) = d(t) * \bar{S}_n(t) \\ \bar{h}(t) = h(t) * \bar{S}_n(t) \end{matrix}\right.

Deep Learning Framework

  • In the 1-D convolution (\(*\)) on Apache MXNet, given input data with shape [batch size, channel, length] :
output[n, i, :] = \sum^{channel}_{j=0} input[n,j,:] \ast weight[i,j,:]

FYI:       \(N_\ast = \lfloor(N-K+2P)/S\rfloor+1\)

（A schematic illustration for a unit of convolution layer)

GW search · Pipeline

Text

Beyond Speed: Generalization and Discovery in GW Detection

  • Our primary goal is not speed but the model's ability to generalize and discover new GW signals, including those beyond the reach of matched filtering techniques and General Relativity (GR).
  • Leveraging our experience in  signal modeling  (MFCNN) and noise modeling (WaveFormer), we are gradually building an offline pipeline capable of searching for signals in complete GW observation data and calculating FARs.

He Wang, et al. PRD 101, 10 (2020): 104003

  • Transform matched-filtering method from frequency domain to time domain.
  • The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

Frequency domain

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)

Time domain

(normalizing)

(matched-filtering)

\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)

\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)

(whitening)

where

\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix} \bar{d}(t) = d(t) * \bar{S}_n(t) \\ \bar{h}(t) = h(t) * \bar{S}_n(t) \end{matrix}\right.

Deep Learning Framework

GW search · Pipeline

Text

Beyond Speed: Generalization and Discovery in GW Detection

  • Our primary goal is not speed but the model's ability to generalize and discover new GW signals, including those beyond the reach of matched filtering techniques and General Relativity (GR).
  • Leveraging our experience in  signal modeling  (MFCNN) and noise modeling (WaveFormer), we are gradually building an offline pipeline capable of searching for signals in complete GW observation data and calculating FARs.

He Wang, et al. PRD 101, 10 (2020): 104003

import mxnet as mx
from mxnet import nd, gluon
from loguru import logger

def MFCNN(fs, T, C, ctx, template_block, margin, learning_rate=0.003):
    logger.success('Loading MFCNN network!')
    net = gluon.nn.Sequential()         
    with net.name_scope():
        net.add(MatchedFilteringLayer(mod=fs*T, fs=fs,
                                      template_H1=template_block[:,:1],
                                      template_L1=template_block[:,-1:]))
        net.add(CutHybridLayer(margin = margin))
        net.add(Conv2D(channels=16, kernel_size=(1, 3), activation='relu'))
        net.add(MaxPool2D(pool_size=(1, 4), strides=2))
        net.add(Conv2D(channels=32, kernel_size=(1, 3), activation='relu'))    
        net.add(MaxPool2D(pool_size=(1, 4), strides=2))
        net.add(Flatten())
        net.add(Dense(32))
        net.add(Activation('relu'))
        net.add(Dense(2))
	# Initialize parameters of all layers
    net.initialize(mx.init.Xavier(magnitude=2.24), ctx=ctx, force_reinit=True)
    return net

The available codes (2019): https://gist.github.com/iphysresearch/a00009c1eede565090dbd29b18ae982c

1 sec duration
35 templates used

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

Network Architecture

  • The WaveFormer, a billion-scale transformer-based model, excels in suppressing realistic noise and recovering injections or GW events, thereby significantly improving data quality.
  • In its application, it treats each overlapping time-domain data subsequence as an individual token, akin to tokenization in natural language processing (NLP).
["This", "is", "a", "sample"]

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Data Preprocessing and Training Strategy

\frac{d-mean}{std} = \frac{h}{std}+\frac{n-mean}{std}

Strain

Whiten

Normalized

 ∼\(10^{−19}\)

 ∼\(10^{2}\)

 ∼\(10^{0}\)

32 s

32 s

merger

\(t_c\) (around GW150914)

\oplus

(Cal network SNR)

Band-pass: [20, 2048] Hz

Patching (tokenized) with size 0.125 s and overlap 50%

[1, 128, 256]

(Standard normalization)

dynamic masking

[1, 16512]

[1, 128, 256]

(PSD\(_i\) from noise)

Band-pass: [20, 2048] Hz

WaveFormer

MSE-Loss\(_i\)

\(std\)

[1, 128, 256]

Noise\(_i\):

Signal\(_i\):

Input\(_i\):

Label\(_i\):

Output\(_i\):

8.0625 s

8.0625 s

Given �=ℎ+�d=h+n, we can normalize �d as follows:

  • Implementations:
    • PSD sampling from real noise.
    • input size: 8.0625 sec
    • fs = ​2048Hz
    • Band-pass: 20~2048Hz
    • Masked loss

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Effect on Realistic Noise

  • Noise level percentile amplitude is significantly reduced, by approximately two orders.
  • Further ASD analysis shows that WaveFormer effectively eliminates both narrowband and broadband spectral information, substantially lowering frequency contributions.
  • Using the Gravity Spy database for glitches with SNR > 10 and confidence > 0.95, results show significant suppression of glitches in real advanced LIGO-Virgo noise.

(Bottom panels: results of glitches)

(Upper panels: results of pure noise)

Time-series and spectrogram example of blip.

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Recovery of Binary Black Holes

  • Overlap and matched-filtering signal-to-noise are calculated to represent phase and amplitude recovery performance.
  • Among the intermediate frequency range (20–200 Hz) that covers rich BBH signal information, the ASD distribution of denoised waveform is evidently consistent with that of target signal.

(Upper panels: Signal amplitude recovery performance

(Bottom panels: Signal phase recovery performance​)

Bacon P. et al.  arXiv: 2205.13513

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

  • These results show that our denoising algorithm outperformed others by capturing the characteristic chirping morphology of BBH evolution, and can denoise signals in realistic detection scenarios without affecting signal characteristics such as phase and amplitude.
  • For the event GW191204_171526, classified as either an NSBH or a low-mass BBH candidate in GWTC-3, the overlap with IMRPhenomXPHM achieved 0.93 and 0.95 on H1 and L1, respectively, which are marked improvements over those achieved by BayesWave and cWB (with overlaps between 0.82–0.86).

GW191204_171526

Recovery of Binary Black Holes

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Search Strategy Overview

  • Firstly, we obtain the denoised output by utilizing Waveformer. Then, triggers are defined and identified by three steps including,

    1. Find Peaks. Locate triggers on a single detector by finding its maximum all local-maximum (0.2s away from neighboring maximum/local-maximum).

  • An search algorithm for GW require that: [cite: 2010.07244]

    1. the same signal is seen in the detectors; (the same signal is seen by time-shifting in single detector)

    2. the same waveform must be present both detectors;

    3. and the signal’s time of arrival must be consistent with the GW travel time between the observatories.

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Search Strategy Overview

  • Firstly, we obtain the denoised output by utilizing Waveformer.
    Then, triggers are defined and identified by three steps including:

    1. Find Peaks. Locate triggers on a single detector by finding its maximum all local-maximum (0.2s away from neighboring maximum/local-maximum).

    2. By constraining triggers that exist on both two detectors, we get VALID triggers. (consist 3~4 segments)

    3. Calculate the cross-correlation of the to-be-evaluated trigger across channels or within a single channel, between its noisy and corresponding denoised segments, as well as between denoised segments themselves.

noisy input segments

denoised output segments

\(\bar{H}\)

\(\bar{L}\)

\({H}\)

\({L}\)

\rho_\text{ranking}

AI

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Significance Estimates

  • Assessed denoising workflow performance by comparing with GWTC-1, GWTC-2, GWTC2.1, and GWTC-3 catalogs and associated data releases.

  • Noted significant divergence in IFAR distribution between our results and those from GWTC and OGC catalogs.

  • Achieved significant IFAR improvement across all 75 reported BBH events, indicating effective suppression of loud terrestrial noise.

    • Example: For low SNR (\(10.8_{-0.4}^{+0.3}\)) event GW200208_130117, obtained an IFAR of 8916 years, surpassing maximum IFAR of <4000 years in other catalogs.

  •  Variability in IFAR improvement linked to the original data's noise nature, including its non-Gaussian, non-stationary characteristics, and different signal recognition strategies by pipelines. 

  • IFAR performance significantly depends on the reduction of non-Gaussian noise near each event.

    • Events with substantial IFAR improvement had misleading non-Gaussian noise effectively eliminated.

    • Events where IFAR underperforms retained non-Gaussian characteristics, possibly due to WaveFormer's inherent systematic errors.

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

  • Developed an AI-based workflow with WaveFormer, combining convolutional neural network and transformer for effective GW noise suppression and hierarchical feature extraction across a wide frequency range.
  • Achieved significant noise suppression and signal recovery performance improvements, including state-of-the-art results on real observational data and BBH events, leading to dramatic data quality improvement and significant IFAR enhancement on 75 reported BBH events.

Text

  • Challenges in Model Interpretability
    • The black-box nature of AI models complicates interpretability, challenging the comparison of AI-generated detection statistics with traditional matched filtering chi-square distributions.
    • Convincing the scientific community of the pipeline's validity and the statistical significance of new discoveries remains difficult despite the model's ability to identify potential gravitational wave signals.

LVK. PRD (2016). arXiv:1602.03839

GW151226

GW151012

GW search · Pipeline

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

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

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

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

（Based on 1912.02762）

Normalizing Flow Model

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

  • Data: target data \(\mathbf{y}\in\mathbb{R}^{15}\) (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.}
p_{\mathrm{y}}(\mathbf{y})
p_{\mathrm{z}}(\mathbf{z})
\mathbf{z}
\mathbf{y}
T
T^{-1}

base density

target density

\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]

Rational Quadratic Neural Spline Flows
(RQ-NSF)

（Based on 1912.02762）

Normalizing Flow Model

  • Bayesian inference, the Holy Grail of gravitational-wave data analysis,
    enables astrophysical interpretation and scientific discoveries.
     

Simulation-Based Inference (SBI)

  • SBI \(\Rightarrow\) Fast and precise parameter estimation.
  • SBI \(\Rightarrow\) TGR / Cosmology / PTA ...

Text

PRL 127, 24 (2021) 241103.

PRL 130, 17 (2023) 171403.

Real-time gravitational wave science with neural posterior estimation

Sampling with prior knowledge for high-dimensional gravitational wave data analysis

He Wang, et al. Big Data Min. Anal. (2021)

PRD 108, 4 (2023): 044029.

Neural Posterior Estimation with Guaranteed Exact Coverage: The Ringdown of GW150914

arXiv:2310.13405, LIGO-P2300306

Cosmological Inference using Gravitational Waves and Normalising Flows

Normalizing Flows as an Avenue to Studying Overlapping Gravitational Wave Signals

DOI: 10.1103/PhysRevLett.130.171402

Parameter Estimation · Scientific Discovery

arXiv:2310.12209

Fast Parameter Inference on Pulsar Timing Arrays with Normalizing Flows

arXiv:2404.14286

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

Space-borne GW Detection: Background

Analyses cannot treat sources independently and sequentially work through a list of candidate detections.

The analysis of scientific data from space-based GW detection differs significantly from ground-based detection:

  • A superposition of overlapping signals ( \(\neq\) isolated event)
  • Observations of more waveform periods over different time scales ( \(\neq\) short-duration signals)
  • Signal-dominated detection ( \(\neq\) noise-dominated)
  • Reliance on more complex techniques for noise assessment
    ( \(\neq\) regular acquisition of signal-free data)
  • The data analyses depend on the simultaneous fitting of complete astrophysical, cosmological, and instrument models to the observed data.
  • The need for this so-called “global fit” was identified as the primary challenge to the data analysis early in the space-borne mission formulation.

(Sec.8.6 LISA Definition Study Report)

Rapid PE for Space-borne GW Detection

  • Global vs. Individual Analysis: While global-fit techniques effectively manage the dense overlapping of signals in space-based GW data, individual pipelines are crucial for detecting unique events.

  • Role of Individual Pipelines: These pipelines act as a pre-processing step, focusing on particular types of sources and diving deeper into the data. They refine the analysis by working on the latest best-fit residuals from the global fit.

  • Case Study - MBHB Mergers: Mergers of MBHBs often exhibit high SNR between \(10^2\) to \(10^3\), appearing as distinct peaks in data time series. 

  • Data curation

    • Model: frequency domain; PhenomD;  TDI-A/E response

    • Input: 1 day length; 15Hz; shape=(2, 3, 2877)

    • Noise: Gaussian stationary from the noise PSD (for training/test) + GB confusion noise (for test)

    • Project: Taiji program

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

Customization for the Taiji scenario: A scalable approach

  • The top section of the illustration shows the solar system barycenter (SSB) and Taiji frames, with two black dashed arrows symbolizing not two separate GW signals, but rather indicating how the sky location and arrival time of the same GW signal take different values in these two frames.

  • The “positive” problem translates the SSB-frame parameters to their Taiji-frame counterparts via a time-dependent mapping \(f_1\), then to the TDI outputs through a time-independent mapping \(f_2\), and an exponential term.

TDI-A

  • These steps can be schematically summarized as:


    where \(\mathcal{T}_\alpha^{A, E}(f)\) is often referred to as the transfer function.

A, E(f)=\sum_\alpha \mathcal{T}_\alpha^{A, E}(f) \tilde{h}_\alpha(f), \quad \alpha \in\{+, \times\}

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

Customization for the Taiji scenario: A scalable approach

  • Consequently, even if the network has only learned the time-dependent relationship between \(\boldsymbol{\theta}_S\) and the TDI response at a specific tref (the 30th day in our case), with the aid of coordinate transformation, it has essentially learned the time-invariant mapping \(f_2\), and can be then generalized to make parameter estimation at any other reference time.
     

  • It is worth noting that our method relies on analytical orbits and
    the time-independence of the coordinate transformation \(f_2\).

  • The top section of the illustration shows the solar system barycenter (SSB) and Taiji frames, with two black dashed arrows symbolizing not two separate GW signals, but rather indicating how the sky location and arrival time of the same GW signal take different values in these two frames.

  • The “positive” problem translates the SSB-frame parameters to their Taiji-frame counterparts via a time-dependent mapping \(f_1\), then to the TDI outputs through a time-independent mapping \(f_2\), and an exponential term.

1 year length

can infer at any other reference time

trained on the 30th day only

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

Unbiased estimation and confidence validation

  • Methodology: Utilization of the Kolmogorov-Smirnov (KS) test to compare one-dimensional distributions generated by our algorithms, ensuring the accuracy of parameter estimation.

  • Empirical Validation: Conducted extensive testing on simulated signals, injecting 1000 waveforms from the prior with added confusion noise and varying reference times between 1 and 365 days.

  • Results: The tests assessed the frequency at which true parameters fell within certain confidence levels, confirming that our credible intervals are well-calibrated and reflect true confidence in the signal parameters.

Computational performance

  • 10000 posterior samples in 2.7 sec

  • The remarkable speed of our method, which outpaces traditional techniques by several orders of magnitude, establishes it as an invaluable tool for preprocessing in global fitting.

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

Multimodality in extrinsic parameters

  • Overview of Findings: Nested sampling results indicate minimal expected multimodality in ecliptic coordinates. However, distinct peaks identified in the time of coalescence (\(t_c\)), labeled as NF-1 (dominant) and NF-2 (subdominant), highlight unique multimodal behavior.

  • Impact on PE: The presence of these peaks affects the posterior distributions of extrinsic parameters, potentially leading to inaccuracies in \(t_c\) and subsequent parameters due to phase term associations and inherent degeneracies.
  • Model Performance: Despite the multimodality, the best-fit values from the NF model closely align with true values within the \(1\sigma\) range for most parameters, and at least \(2\sigma\) for others.
  • Comparative Analysis: The ML pipeline tends to produce broader posteriors compared to the Bayesian nested sampling approach. 

（NF = Normalizing Flow model）

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

Content

  • GW Astronomy · GW Data Analysis
  • GW search · Pipeline
    • 1. MFCNN (feature extraction for detection)
    • ​2. WaveFormer (denoising for detection)
  • Parameter Estimation · Scientific Discovery
    • 3. Rapid PE for Space-borne GW Detection
  • ​Key Takeaways & Look Forwards

Key Takeaways & Look Forwards

Text

AI vs Classical Methods

  • In strict sense, at present AI methods have not yet become the perfect alternative to traditional methods (like MF, MCMC, etc.).
    • There is a pressing need for the theoretical refinement of ML applications in GW statistics, aiming to bridge current gaps and enhance model reliability.
    • How can we address the issue of strong or unacceptable biases that occur when outputs from AI models are used jointly or in combination to measure properties of a population, sub-population, or ensemble?
      (also addressed by 2405.18095)

Alfaidi & Messerger.  arXiv:2402.04589

Our result (preliminary)

Menéndez-Vázquez A, et al. PRD 2021

PyGWB result

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

Text

Statistics

\times N
\times N

Statistics

AI vs Classical Methods

  • In strict sense, at present AI methods have not yet become the perfect alternative to traditional methods (like MF, MCMC, etc.).
    • There is a pressing need for the theoretical refinement of ML applications in GW statistics, aiming to bridge current gaps and enhance model reliability.
    • How can we address the issue of strong or unacceptable biases that occur when outputs from AI models are used jointly or in combination to measure properties of a population, sub-population, or ensemble?
      (also addressed by 2405.18095)
  • How can we construct a gravitational wave statistical theory based on machine learning methods to achieve robust theoretical guarantees for detection statistics and statistical inference?
    • \(\rightarrow\) Is the goal to achieve results that are entirely consistent with the statistical properties of traditional methods, or
    • \(\rightarrow\) to quantify and calibrate any results provided by AI within the statistical theory framework?

Key Takeaways & Look Forwards

Text

Calibration​ Analysis on Denoised / Residual Data

  • Denoised data
    • ​Non-Gaussian and non-stationary.
    • The SNR is evidently high, but it cannot be utilized for parameter estimation. Because:
      • Background noise is also non-Gaussian.
      • Assessing the unbiasedness of the posterior distribution will be very challenging.
    • Is it possible to calibrate or validate the denoised data simultaneously while training the denoiser?
  • Residual Data (“Global-fit” challenge for LISA)
    • Iteratively using exact waveforms is not practical in terms of timeliness, while using approximate waveforms inevitably introduces systematic errors in the residual data during each iteration.
    • Is it possible to calibrate the residual data simultaneously during each iteration of the statistical inference?

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

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

Key Takeaways & Look Forwards

for _ in range(num_of_audiences):
    print('Thank you for your attention! 🙏')

This silde: https://slides.com/iphysresearch/2024august_uoa

Text

Calibration​ Analysis on Denoised / Residual Data

  • Denoised data
    • ​Non-Gaussian and non-stationary.
    • The SNR is evidently high, but it cannot be utilized for parameter estimation. Because:
      • Background noise is also non-Gaussian.
      • Assessing the unbiasedness of the posterior distribution will be very challenging.
    • Is it possible to calibrate or validate the denoised data simultaneously while training the denoiser?
  • Residual Data (“Global-fit” challenge for LISA)
    • Iteratively using exact waveforms is not practical in terms of timeliness, while using approximate waveforms inevitably introduces systematic errors in the residual data during each iteration.
    • Is it possible to calibrate the residual data simultaneously during each iteration of the statistical inference?

He Wang et al 2024 Mach. Learn.: Sci. Technol. 5 015046

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

Key Takeaways & Look Forwards

Parameter estimation · Scientific discovery

Text

Exploring Stochastic Gravitational Wave Background with AI

  • Utilizing AI for parameter estimation in the stochastic gravitational wave background (SGWB) presents a fascinating blend of rich theoretical content and the potential for optimizing current data processing methods.
  • While still preliminary and ongoing, our work shows promising results for high SNR SGWB scenarios, where AI-based posterior probabilities are notably more precise and narrower compared to traditional cross-correlation methods used in PyGWB.
\Omega_{\mathrm{GW}}(f)=\Omega_{\mathrm{ref}}\left(\frac{f}{f_{\mathrm{ref}}}\right)^\alpha
\Omega_{\mathrm{ref}}=10^{-6.1}

Our result (preliminary)

Parameter estimation · Scientific discovery

Text

Exploring Stochastic Gravitational Wave Background with AI

  • Performance saturation is observed between SNR levels of \(10^{-6}\) to \(10^{-7}\), indicating a plateau in model effectiveness in low SNR conditions.
  • Unlike PyGWB, which can accumulate cross-correlation data from SGWB to further constrain the power spectrum, AI model outputs do not readily provide statistically meaningful information for aggregation. Multiplying posterior probabilities from multiple segments leads to ambiguous, and potentially biased, results due to the lack of statistically significant fluctuations across different posterior distributions.

Abbott R, et al. PRD 104, 2 (2021): 022004.

PyGWB result

Our result (preliminary)

\Omega_{\mathrm{GW}}(f)=\Omega_{\mathrm{ref}}\left(\frac{f}{f_{\mathrm{ref}}}\right)^\alpha