Pre-localization of Massive Black Hole Binaries in the Millihertz Band

The space-borne gravitational-wave (GW) detectors will open a new mass and redshift regime, allowing us to observe massive black hole binaries (MBHBs) throughout the Universe. A subset of these systems is expected to produce electromagnetic (EM) counterparts, offering a unique opportunity to follow the continuous evolution of massive black holes through joint GW and EM observations. Realizing this potential, however, requires low-latency, high-throughput data-analysis pipelines that can extract reliable source parameters and sky localizations from space-borne data streams fast enough to trigger EM follow-up. In this work we develop a fast, normalising flow-based inference pipeline designed for early-warning analysis of MBHB signals in a TianQin-like configuration. Our method combines a learned embedding of the detector time series with a neural spline flow (NSF) to perform amortized Bayesian inference, producing posterior samples for the main source parameters in roughly one minute per event. For a representative MBHB whose merger occurs $\sim 15$ minutes after the end of the analyzed GW observation, the pipeline achieves pre-merger sky localizations of order $\sim 20~\mathrm{deg}^2$, recovers the same number of sky modes as a reference parallel-tempered Markov chain Monte Carlo (PTMCMC) analysis, and yields parameter uncertainties of comparable scale, while still operating within a practically useful pre-merger warning window. These results demonstrate that NSF-based inference can deliver accurate, near-real-time parameter estimation for space-borne MBHB GW signals, and that the resulting early-warning localizations are sufficiently precise to make rapid EM follow-up.