Deep Learning Applications in Gravitational Wave Data Analysis: From Discovery to Characterization

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 KAGRA collaborations / Taiji collaborations

Contents

Gravitational waves (GW) are a strong field effect in General Relativity, ripples in the fabric of spacetime caused by accelerating massive objects.

Gravitational Wave Astronomy

Challenge and Methodology: Detecting Signals in GW Data

GW Data Characteristics

LIGO-VIRGO-KAGRA

LISA Project

• Noise: non-Gaussian and non-stationary

• Signal challenges:

  • (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.: 10⁴ of GBs, 10~10² of SMBHs, and 10~10³ of EMRIs, etc.) received during the mission's observational run.

Matched Filtering Techniques (匹配滤波方法)

• 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).

Statistical Approaches

Core Insight from Computer Vision

• Direct approach from Computer Vision (CV) to GW signal processing: pixel point \(\Rightarrow\) sampling point.
• The CNN framework treats time series data similar to images, where each sampling point represents a feature to learn.

Performance Analysis

• Convolutional neural networks (CNN) can achieve comparable performance to Matched Filtering under Gaussian stationary noise.
• CNNs significantly outperform traditional methods in terms of execution speed (with GPU support).
• Modern architectures show improved robustness against non-Gaussian noise transients (glitches).

Pioneering Research Publications

PRL, 2018, 120(14): 141103.

PRD, 2018, 97(4): 044039.

CNN for GW Detection: Pioneering Approaches

Universal Approximation Theorem: Existence Theorem

• Neural networks with sufficient hidden layers can approximate any continuous function on compact subsets of \(\mathbb{R}^n\).
• For GW detection, this means CNNs can theoretically learn the optimal detection statistics without explicit physical modeling.
• The expressive power of deep neural networks enables capturing complex patterns in non-Gaussian, non-stationary noise.
• Ref: Cybenko, G. (1989), Hornik et al. (1989)

Beyond Speed: Generalization and Explainability

• Improving AI explainability reveals deep connections between CNN architectures and matched filtering techniques.
• Matched-filtering (cross-correlation with templates) can be interpreted as a convolutional layer with predefined kernels.
• In practice, we use matched filters as an essential component of feature extraction in CNNs for GW detection.

Convolutional Neural Network (ConvNet or CNN)

feature extraction

classifier

Matched-filtering Convolutional Neural Network (MFCNN)

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

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

CNN for GW Detection: Theoretical Foundation

>> Is it matched-filtering ?
>> Wait, It can be matched-filtering!

Universal Approximation Theorem: Existence Theorem

• Neural networks with sufficient hidden layers can approximate any continuous function on compact subsets of \(\mathbb{R}^n\).
• For GW detection, this means CNNs can theoretically learn the optimal detection statistics without explicit physical modeling.
• The expressive power of deep neural networks enables capturing complex patterns in non-Gaussian, non-stationary noise.
• Ref: Cybenko, G. (1989), Hornik et al. (1989)

Beyond Speed: Generalization and Explainability

• Improving AI explainability reveals deep connections between CNN architectures and matched filtering techniques.
• Matched-filtering (cross-correlation with templates) can be interpreted as a convolutional layer with predefined kernels.
• In practice, we use matched filters as an essential component of feature extraction in CNNs for GW detection.

Convolutional Neural Network (ConvNet or CNN)

Matched-filtering Convolutional Neural Network (MFCNN)

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

CNN for GW Detection: Theoretical Foundation

GW150914

GW150914

CNN for GW Detection: Theoretical Foundation

• Transform matched-filtering method from frequency domain to time domain.

• The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

where

Deep Learning Framework

• In the 1-D convolution (\(*\)) on Apache MXNet, given input data with shape [batch size, channel, length] :

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

（A schematic illustration for a unit of convolution layer)

Time Domain

(matched-filtering)

(normalizing)

(whitening)

CNN for GW Detection: Theoretical Foundation

• Transform matched-filtering method from frequency domain to time domain.

• The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

where

Deep Learning Framework

Time Domain

(matched-filtering)

(normalizing)

(whitening)

CNN for GW Detection: Theoretical Foundation

1 sec duration

35 templates used

Explainable AI Approach

• Implements matched filtering operations through custom convolutional layers
• Makes the network more interpretable by embedding domain knowledge
• Connects traditional signal processing with deep learning
• Outperforms standard CNNs in both accuracy and efficiency

Matched-filtering Convolutional Neural Network (MFCNN)

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

• Gravitational wave signal search algorithm benchmark (MLGWSC-1)
• Dataset-4: Sampled from O3a real gravitational wave observation data

First Benchmark for GW Detection Algorithms

Benchmark Results

Publications

arXiv:2501.13846 [gr-qc]

Interpretability Challenges: Comparing Detection Statistics

• 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.

AI Model Denoising

Our Model's Detection Statistics

LVK Official Detection Statistics

Signal denoising visualization using our deep learning model (Transformer-based)

He Wang et al 2024 MLST 5 015046

Detection statistics from our AI model showing O1 events

He Wang et al 2024 MLST 5 015046

GW151226

GW151012

Official detection statistics from LVK collaboration

LVK. PRD (2016). arXiv:1602.03839

Interpretability Challenges: Discoveries vs. Validation

arXiv:2407.07820 [gr-qc]

Recent AI Discoveries & Validation Hurdles:

• A recent study (arXiv:2407.07820) demonstrates how a ResNet-based (CNN) architecture with careful signal search strategy and post-processing can identify 8 new potential gravitational wave events from LIGO O3 data.
• The absence of these events in traditional PyCBC results raises questions: could adjustments to rate priors and p_astro parameters in signal models help traditional pipelines detect these candidates (if they are real GW events)?
• The ideal approach combines multiple diverse pipelines working in parallel to ensure comprehensive detection (requiring interpretable models) and using evidence-based detection statistics while simultaneously optimizing both real signal population (p_astro) and noise model (likelihood) fits.

Search

PE

Rate

Interpretability Challenges: Discoveries vs. Validation

Parameter Estimation Challenges with AI Models:

• In parameter estimation, AI models' lack of interpretability requires substantial additional scientific validation to ensure credibility and acceptance of results.
• Parameter distributions from AI models often lack robustness across different noise realizations and are difficult to calibrate against established methods.
• Scientific papers using AI methods must dedicate significant space to validation procedures, comparing against traditional methods and demonstrating reliability across multiple test cases.

arXiv:2404.14286

Phys. Rev. D 109, 123547 (2024)

The Next Frontier:
LLMs for Gravitational Wave Data Analysis

Given the interpretability challenges we've explored,
how might we advance GW detection and parameter estimation while maintaining scientific rigor?

The Next Frontier:
LLMs for Gravitational Wave Data Analysis

Given the interpretability challenges we've explored, how might we advance GW detection and parameter estimation while maintaining scientific rigor?

Automatic and Evolutionary Algorithm Heuristics for GW Detection using LLMs

A promising new approach combining the power of large language models with evolutionary algorithms to create interpretable, adaptive detection systems

The Rise of LLMs: How Code Training Transformed AI Capabilities

Evolution of GPT Capabilities

A careful examination of GPT-3.5's capabilities reveals the origins of its emergent abilities:

• Original GPT-3 gained generative abilities, world knowledge, and in-context learning through pretraining
• Instruction-tuned models developed the ability to follow directions and generalize to unseen tasks
• Code-trained models (code-davinci-002) acquired code comprehension
• The ability to perform complex reasoning likely emerged as a byproduct of code training

GPT-3.5 series [Source: University of Edinburgh, Allen Institute for AI]

GPT-3 (2020)

ChatGPT (2022)

Magic: Code + Text

Recent research demonstrates that LLMs can solve complex optimization problems through carefully engineered prompts. DeepMind's OPRO (Optimization by PROmpting) approach showcases how LLMs can generate increasingly refined solutions through iterative prompting techniques.

OPRO: Optimization by PROmpting

Example: Least squares optimization through prompt engineering

arXiv:2309.03409 [cs.NE]

Two Directions of LLM-based Optimization

arXiv:2405.10098 [cs.NE]

The Optimization Potential of Large Language Models

LLMs can generate high-quality solutions to optimization problems without specialized training

Theoretical Understanding of LLMs' Emergent Abilities

The Interpolation Theory

LLMs' ability to generate novel responses from few examples is increasingly understood as manifold interpolation rather than mere memorization:

• LLMs learn a continuous semantic manifold of language during pre-training
• Few-shot examples serve as anchor points in this high-dimensional space
• The model interpolates between examples to generate responses for novel inputs
• This enables coherent generalization beyond the training distribution
• The quality of interpolation improves with model scale and training data breadth

The theory suggests that in-context learning is not "learning" in the traditional sense, but rather a form of implicit conditioning on the manifold of learned representations.

Representation Space Interpolation

Real-world Case: FunSearch (Nature, 2023)

• Google DeepMind's FunSearch system pairs LLMs with evaluators in an evolutionary process
• Discovered new mathematical knowledge for the cap set problem in combinatorics, improving on best known bounds
• Also created novel algorithms for online bin packing that outperform traditional methods
• Demonstrates LLMs can make verifiable scientific discoveries beyond their training data

Theoretical Understanding of LLMs' Emergent Abilities

The Interpolation Theory

LLMs' ability to generate novel responses from few examples is increasingly understood as manifold interpolation rather than mere memorization:

• LLMs learn a continuous semantic manifold of language during pre-training
• Few-shot examples serve as anchor points in this high-dimensional space
• The model interpolates between examples to generate responses for novel inputs
• This enables coherent generalization beyond the training distribution
• The quality of interpolation improves with model scale and training data breadth

The theory suggests that in-context learning is not "learning" in the traditional sense, but rather a form of implicit conditioning on the manifold of learned representations.

Representation Space Interpolation

Key Literature

• Wei et al. (2022) - "Emergent Abilities of Large Language Models" arXiv:2206.07682
• Akyürek et al. (2022) - "What learning algorithm is in-context learning?" arXiv:2211.15661
• Min et al. (2022) - "Rethinking the Role of Demonstrations" arXiv:2202.12837
• Xie et al. (2022) - "An Explanation of In-context Learning as Implicit Bayesian Inference" arXiv:2111.02080

Theoretical Understanding of LLMs' Emergent Abilities

The Interpolation Theory

LLMs' ability to generate novel responses from few examples is increasingly understood as manifold interpolation rather than mere memorization:

• LLMs learn a continuous semantic manifold of language during pre-training
• Few-shot examples serve as anchor points in this high-dimensional space
• The model interpolates between examples to generate responses for novel inputs
• This enables coherent generalization beyond the training distribution
• The quality of interpolation improves with model scale and training data breadth

The theory suggests that in-context learning is not "learning" in the traditional sense, but rather a form of implicit conditioning on the manifold of learned representations.

Representation Space Interpolation

Key Literature

• Wei et al. (2022) - "Emergent Abilities of Large Language Models" arXiv:2206.07682
• Akyürek et al. (2022) - "What learning algorithm is in-context learning?" arXiv:2211.15661
• Min et al. (2022) - "Rethinking the Role of Demonstrations" arXiv:2202.12837
• Xie et al. (2022) - "An Explanation of In-context Learning as Implicit Bayesian Inference" arXiv:2111.02080

Theoretical Understanding of LLMs' Emergent Abilities

The Interpolation Theory

LLMs' ability to generate novel responses from few examples is increasingly understood as manifold interpolation rather than mere memorization:

• LLMs learn a continuous semantic manifold of language during pre-training
• Few-shot examples serve as anchor points in this high-dimensional space
• The model interpolates between examples to generate responses for novel inputs
• This enables coherent generalization beyond the training distribution
• The quality of interpolation improves with model scale and training data breadth

The theory suggests that in-context learning is not "learning" in the traditional sense, but rather a form of implicit conditioning on the manifold of learned representations.

Representation Space Interpolation

Key Literature on Manifold Interpolation

• Raventos et al. (2023) - "In-Context Learning Dynamics with Manifold Identification" arXiv:2305.12104
• Garg et al. (2022) - "What Can Transformers Learn In-Context? A Case Study of Simple Function Classes" arXiv:2208.01066
• Dai et al. (2022) - "Why Can GPT Learn In-Context?" arXiv:2212.10559
• Xie et al. (2022) - "An Explanation of In-context Learning as Implicit Bayesian Inference" arXiv:2111.02080

https://www.lesswrong.com/posts/GADJFwHzNZKg2Ndti/have-llms-generated-novel-insights

https://gowrishankar.info/blog/deep-learning-is-not-as-impressive-as-you-think-its-mere-interpolation/

REWIRING AGI—NEUROSCIENCE IS ALL YOU NEED

What is test-time scaling?

Why LLMs can do the inference/optimation?
How about the theory? (check: 2410.14716)

Why we need MCTS?
Why and How is Evoluation theory in Opt area?

Add computational complexity analysis

借用流浪地球的台词？

借用流浪地球的台词？

Drawbacks and limitations：hard control for opt direction(when to balance between exploration and exploitation)；sensitive to prompt template / LLM version; hard to define the search space for the unknown solution when problem is complicated;

好好先review一下：eccentricity using DINGO; AreaGW

自己实验的OPRO效果

好好先review一下：eccentricity using DINGO; AreaGW

逐层递进深刻的reflection

Algorithmic Synergy: MCTS, Evolution & LLM Agents

Monte Carlo Tree Search (MCTS)

• Efficiently explores high-dimensional spaces
• Balances exploration and exploitation
• Provides strong theoretical guarantees
• Excels in complex sequential decision-making

Evolutionary Algorithms

• Enables gradient-free global optimization
• Naturally handles multi-objective problems
• Provides diverse solution candidates
• Robust to noisy objective functions

LLM Agents

• Processes and generates domain-specific text
• Understands and generates code
• Reasons through complex problem spaces
• Adapts strategies based on context

Algorithmic Synergy: MCTS, Evolution & LLM Agents

Proposed framework integrating MCTS decision-making, self-evolutionary optimization, and LLM agent guidance for gravitational wave signal search

With route/short/long-term reflection：《Thinking, Fast and Slow》

• deepseek-R1 for reflection generation
• gpt-4o-2024-11-20 /claude-3-7-sonnet-20250219 for code generation

Preliminary Results (February 2025)

Algorithmic Exploration：Seed Function

Function Role in Framework

• Serves as the initial solution that will be evolved and optimized by the framework
• Provides baseline GW signal detection capability
• Acts as the starting point for MCTS exploration
• Establishes the structure that LLM agents will modify

Pipeline Workflow

1. Conditions raw detector data (whitening)
2. Computes time-frequency metrics
3. Identifies peaks above background
4. Returns event candidates with timestamps

Input: H1 and L1 detector strains, time array | Output: Event times, significance values, and time uncertainties

Preliminary Results (February 2025)

Algorithmic Exploration：LLM Prompt Engineering

Prompt Structure for Algorithm Evolution

This template guides the LLM to generate optimized gravitational wave detection algorithms by learning from comparative examples.

Key Components:

• Expert role establishment
• Example pair analysis (worse/better algorithm)
• Reflection on improvements
• Targeted new algorithm generation
• Strict output format enforcement

One Prompt Template for MLGWSC1 Algorithm Synthesis

Preliminary Results (February 2025)

MLGWSC1 preliminary 结果

MLGWSC1: Algorithm Evolutionary Tree Visualization

Tree-based representation of our framework's exploration path, where each node represents a unique algorithm variant generated during the optimization process

Node color intensity: Algorithm performance level | Connections: Algorithmic modifications | Tree depth: Iteration sequence

Preliminary Results (February 2025)

MLGWSC1 Benchmark: Optimization Performance Results

Preliminary Results (February 2025)

Optimization Progress & Algorithm Diversity

Sensitivity vs False Alarm Rate

Optimization Target: Maximizing Area Under Curve (AUC) in the 10-100Hz frequency range, balancing detection sensitivity and false alarm rates across algorithm generations

Optimization Target: Maximizing Area Under Curve (AUC) in the 10-100Hz frequency range, balancing detection sensitivity and false alarm rates across algorithm generations

Preliminary Results (February 2025)

This pipeline combines adaptive PSD whitening and multi-band spectral coherence computation with a noise floor-aware peak detection and a non-linear timing uncertainty model to enhance gravitational wave signal detection accuracy and robustness.

Integrate asymmetric PSD whitening, extended STFT overlap optimization, chirp-enhanced prominence scaling, multi-channel noise floor refinement, and dynamic timing calibration for improved gravitational wave signal detection.

Optimization Target: Maximizing Area Under Curve (AUC) in the 10-100Hz frequency range, balancing detection sensitivity and false alarm rates across algorithm generations

Optimization Progress & Algorithm Diversity

MLGWSC1 Benchmark: Optimization Performance Results

Preliminary Results (February 2025)

The framework (LLMs) can effectively optimize complex algorithms and guide iterative development along specified optimization directions, achieving targeted performance improvements in GW detection 

MLGWSC1 Benchmark: Optimization Performance Results

Preliminary Results (February 2025)

Sensitivity vs False Alarm Rate

PyCBC

CNN-like

Simple non-linear filter

Key Finding: Our framework demonstrates potential to optimize highly interpretable and scalable non-linear algorithm pipelines that achieve performance comparable to traditional matched filtering techniques.

The Evolution of Scientific Analysis Paradigms

Traditional Physics Approach

Input

Human-Designed Algorithm

(Based on human insight)

Output

Example: Matched Filtering

Black-Box AI Approach

Input

AI Model

(Low interpretability)

Output

Examples: CNN, AlphaGo

Interpretable AI Approach

Input

Optimized
Algorithm

(High interpretability)

Output

Example: OURS (on-going)

The Future: Combining traditional physics knowledge with LLM-optimized algorithms for transparent, reliable scientific discovery

Data/
Experience

Data/
Experience

AI Model

Data/
Experience

Summary: AI for Gravitational Wave Science

Key Insights from Our Journey

• Deep learning methods have transformed GW data analysis, enabling detection capabilities that complement traditional approaches
• Evolution from simple CNN architectures to sophisticated frameworks that leverage domain knowledge
• LLM-guided algorithmic optimization demonstrates potential for creating high-performance, interpretable methods
• Balancing sensitivity and false alarm rates remains a key challenge
• Benchmark results validate the potential of AI-driven approaches in scientific discovery

The Critical Role of Interpretability

Algorithm interpretability provides multiple essential benefits:

• Scientific Understanding: Reveals unique characteristics of different model architectures and their decision processes
• Algorithm Interpolation: Enables meaningful combination of different approaches by understanding their complementary strengths
• Result Controllability: Provides confidence in outcomes and minimizes unexplained behaviors
• Model Calibration: Allows fine-tuning of algorithms based on physical understanding rather than black-box optimization

The future of gravitational wave science lies at the intersection of traditional physics-inspired methods and interpretable AI approaches, creating a new paradigm for reliable scientific discovery.