EMRI Search and Inference within the LISA Global Fit — Part II
National University of Singapore · 14 July 2026

ICTP-AP, University of Chinese Academy of Sciences (UCAS)
Taiji Laboratory for Gravitational Wave Universe, UCAS

In collaboration with Dingkai Zhang, South China Normal University

(in prep.)

He Wang

One learned foreground model for calibrated noise reconstruction and foreground-aware source inference

Learning a Robust Foreground Models for Multi-Source GW Inference

The millihertz data stream is signal-dominated, not noise-dominated.

Tens of millions of Galactic binaries overwhelm the instrumental floor across the band where many high-value sources live.

One-year Taiji simulation

  • \(\sim 4.5\times10^7\) Galactic binaries
  • numerical Taiji orbit
  • unequal-arm TDI-2.0
  • frequency-domain \(A/E\) channels

Why is this difficult?

  • Confusion dominates: \(\sim0.3\)–\(10\) mHz
  • No signal-free frequency region

  • Foreground and instrumental noise cannot be separated directly

There is no clean off-source measurement of the instrumental PSD.

How can we estimate the instrumental PSD

when no frequency region contains only noise?

Credit: Dingkai Zhang

\(\widehat S_{\rm res}(f)\neq S_{\rm inst}(f)\)

  • foreground power contaminates the residual periodogram
  • the inferred PSD is systematically distorted
  • frequency-dependent likelihood weights become incorrect

Galactic foreground biases instrumental PSD estimation.

Signal confusion creates two coupled inference failures

Foreground contamination biases the noise model and leaves structured residuals that degrade downstream source inference.

A contaminated residual means both a distorted PSD and a misspecified source likelihood.

Noise characterization fails

\(r(f;\theta)=d(f)-h(f;\theta)=g_{\rm GB}(f)+n(f)\)

\(r(f;\theta)\not\sim\mathcal N\!\left(0,S_{\rm inst}(f)\right)\)

  • the residual contains structured foreground power
  • the assumed Gaussian-noise likelihood is misspecified
  • posteriors can broaden, shift, or become overconfident

Target-source model is misspecified

Littenberg & Cornish (2023)

Foreground residuals degrade source inference.

Deng, Babak, and Marsat (2025)

For noise calibration and target inference, the aggregate foreground may be more important than the identity of every source.

The global fit solves the problem—but at a price

A global fit resolves the coupling in principle, but does so by explicitly modeling individual sources.

Scientific Question

Do we need the identity of every Galactic binary

to infer the instrumental noise

or validate one target source?

Katz, Karnesis, Korsakova, Gair, and Stergioulas (2025)

What the global fit provides

  • individual source catalogs
  • physical noise posterior
  • joint Bayesian consistency
  • interpretable residuals

What it costs

  • trans-dimensional sampling
  • repeated waveform generation
  • blocked iterative convergence
  • computational cost scales with source number

One learned foreground model, three uses.

  1. Learn
    population-conditioned training
    \(\rightarrow\) frequency-bin–shared foreground network
  2. Characterize
    foreground-subtracted residual
    \(\rightarrow\) unbiased \(A_{\rm acc}, A_{\rm oms}\) + bounded distortion
  3. Deploy in-loop
    cached subtraction inside MBHB MCMC
    \(\rightarrow\) oracle-level posteriors

Our key design choice: learn the aggregate foreground

The network predicts the total Galactic foreground, which is then reused for noise characterization and source inference.

左侧图

四个块：

  1. GB catalog
  2. Instrument noise
  3. MBHB / EMRI / other targets
  4. Residual validation

箭头循环：

GB subtraction
$\rightarrow$ noise update
$\rightarrow$ target-source inference
$\rightarrow$ residual
$\rightarrow$ GB update

Katz, Karnesis, Korsakova, Gair, and Stergioulas (2025)

(adapted)

Aggregate

Foreground

learned by

Pipeline overview

No catalog in, no catalog out.

Aggregate foreground is the prediction target.

Population-conditioned simulation

The network learns a population—not a catalog

Every training sample is a new foreground realization, preventing catalog memorization.

TDC II catalog

\(\rightarrow\) sample source parameters and counts

\(\rightarrow\) GBGPU + Taiji response
\(\rightarrow\) TDI-2.0 A/E segment

Conditional normalizing flow:

  • Target: \(p(\theta_{\rm GB}\mid \log_{10}f)\)

Key choices:

  • full catalog, no SNR split

No realization is ever repeated.

Population validation near 1 mHz

p(\theta_{\rm GB}\mid \log_{10}f)
\log_{10}f
\theta_{\rm GB}

Conditional normalizing flow

One frequency-bin–shared network

One shared network covers the full analysis band

Local spectral morphology is sufficiently universal for one shared network to operate across the entire band.

Input representation

\(x\in\mathbb{R}^{4\times512}\)

Local 512-bin window in A/E channels

Why one shared network?

  • local morphology is translation-invariant
  • dense and sparse regions share the same representation
  • frequency enters only through conditioning

Sliding inference

  • overlapping windows
  • crop borders
  • stitch predictions

Key choices:

  • random source counts: \(N_{\rm ref}(f)\pm2\sqrt{N_{\rm ref}(f)}\)
  • bleed padding near segment edges

Shared morphology enables a shared network.

The architecture is simple; the training objective makes the residual usable

The loss is designed for the downstream likelihood, not just foreground reconstruction.

Calibrate the residual, not just the prediction.

Downstream-aware objective

\(\mathcal{L} = \mathcal{L}_{\rm MSE} + \lambda\mathcal L_{\rm IS},\qquad\lambda=0.15\)

\(\mathcal{L}_{\rm IS}=\frac{1}{M}\sum_{c,j}\left[\frac{R_{c,j}}{T_{c,j}}-\ln\frac{R_{c,j}}{T_{c,j}}-1\right]\)

where

  • \(R_{c,j}=\left\langle |d_c-\hat g_c|^2\right\rangle_j\) is block-averaged residual power
  • \(T_{c,j}=\left\langle |n_c|^2\right\rangle_j\) is block-averaged injected noise power
  • \(j\): eight coarse 64-bin sub-bands

Loss landscape

One shared network generalizes across the full Galactic foreground

No retraining. No per-band tuning. One model everywhere.

The goal is not to identify every binary, but to recover a noise-dominated residual.

Subtraction residual

Reconstructed GB foreground

Input GB foreground

From 512-bin predictions to a continuous full-band residual

Sliding 512-bin windows are edge-cropped and stitched so that every frequency bin receives one reliable prediction.

Local predictions become one continuous residual across the full analysis band.

Full-band inference

  • 512-bin sliding windows
  • 10% edge crop: 51 bins per side
  • stride: 410 bins
  • reflection padding at the band boundaries

 

\(\hat g_c(f)=\operatorname{stitch}\left[M_{\rm GB}(d_c^{(w)})\right] \)

\(r_c(f)=d_c(f)-\hat g_c(f)\)

Instrumental PSD

Input GB foreground

Smoothed Residual

Only a small broadband mismatch remains.

One learned foreground model, three uses.

  1. Learn
    population-conditioned training
    \(\rightarrow\) frequency-bin–shared foreground network
  2. Characterize
    foreground-subtracted residual
    \(\rightarrow\) unbiased \(A_{\rm acc}, A_{\rm oms}\) + bounded distortion
  3. Deploy in-loop
    cached subtraction inside MBHB MCMC
    \(\rightarrow\) oracle-level posteriors

The learned foreground enters Bayesian inference

The same foreground model is reused for both noise characterization and source inference.

左侧图

四个块：

  1. GB catalog
  2. Instrument noise
  3. MBHB / EMRI / other targets
  4. Residual validation

箭头循环：

GB subtraction
$\rightarrow$ noise update
$\rightarrow$ target-source inference
$\rightarrow$ residual
$\rightarrow$ GB update

Katz, Karnesis, Korsakova, Gair, and Stergioulas (2025)

(adapted)

Learned

Foreground

Model

reused in

Pipeline overview

The learned foreground is reused—not retrained—inside downstream inference.

The learned foreground enters the likelihood

Every MCMC proposal is cleaned before evaluating the Whittle likelihood.

The sampler never fits the raw data — it fits the foreground-cleaned residual.

Model in the loop: The foreground prediction is refreshed every 15 iterations during burn-in and then frozen.

Every MCMC proposal is evaluated on the cleaned residual.

Foreground-cleaned residual

Foreground-aware inference removes bias from both the noise and the source posterior

Correcting the foreground first restores both instrumental-noise calibration and MBHB parameter estimation.

??

Noise characterization becomes unbiased

  • recovered \(A_{\rm acc}\) and \(A_{\rm oms}\)
  • posterior centered on the truth
  • spline absorbs remaining smooth mismatch
  • no systematic PSD inflation

MBHB posterior becomes unbiased

  • intrinsic parameters: correct mode, slightly broader posterior
  • extrinsic parameters: nearly oracle-equivalent
  • residual foreground uncertainty is propagated, not hidden

Calibrating learned systematic uncertainty

The learned foreground is not assumed perfect—its remaining error is explicitly calibrated and propagated into the posterior.

Existing uncertainty

  • detector noise
  • finite observation
  • finite measurement precision

 

New uncertainty from learning

  • imperfect foreground prediction

  • residual foreground mismatch

                       ↓

      explicitly calibrated

 

Posterior interpretation

  • bias largely removed
  • remaining foreground uncertainty propagates into credible intervals
  • intrinsic parameters become slightly broader
  • no hidden over-confidence

Our goal is not to eliminate uncertainty, but to separate, calibrate, and propagate it honestly. 

The maximum-likelihood waveform matches the injection.

The learned foreground closely reproduces the aggregate Galactic signal, while leaving only instrument-like residuals.

Toward weaker, longer-lived targets: the EMRI-like regime

 

Bonus​: toward EMRI-like targets

EMRIs are the natural next target because foreground bias accumulates over many cycles.

EMRI-like target

  • months to years
  • \(10^4\)–\(10^5\) cycles
  • low instantaneous SNR
  • foreground bias accumulates coherently

 

左右对比 MBHB vs EMRI-like。不要放复杂 EMRI 波形；用 conceptual timeline 更清楚。

MBHB is the first hard test; EMRI is the real stress test.

最好有EMRI的Result corner图

Needed extensions:

  • distributional \(\hat g\)
  • pseudo-marginal / variance-inflated likelihood
  • time–frequency conditioning
  • phase-coherent validation

Conclusions and outlook

Machine learning provides a practical and statistically honest bridge toward scalable LISA/Taiji global-fit inference.

✓ Learn the aggregate Galactic foreground, not individual binaries

✓ Recover unbiased instrumental-noise calibration

✓ Restore MBHB posterior consistency without source catalogs


Foreground estimation becomes an inference primitive.

 

 

 

 

 

 

Main contributions

AI learns NOT individual GB identities

↓

BUT a morphology → foreground map

↓

usable residuals for downstream inference

Different question, complementary answer.

What AI actually learned

Not every Galactic binary must be identified before precision inference becomes possible.

Conclusions and outlook

Machine learning provides a practical and statistically honest bridge toward scalable LISA/Taiji global-fit inference.

  • The systematic error of the learned component becomes a target of Bayesian calibration.​

Calibratable machine learning

  1. Gee-Moo for TDCII (with Jianan Zhang, ICTP-AP)
  2. MDP for global-fit (Frequentist vs Bayesian ...)
  3. ...

Ongoing work

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Stachurski+ (2024)

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