Probabilistic Interpolation of Sagittarius A*'s Multi-Wavelength Light Curves Using Diffusion Models

TL;DR

We address reconstructing latent signals $X^{(i)}(t)$ from sparse observations ${Y^{(i)}(t_k)}$ of Sgr A* across four bands, using a diffusion-based continuous-time framework (CSPD) for probabilistic interpolation. The approach is trained on a large simulated dataset that mimics realistic cadences and noise and is compared against a multi-output Gaussian Process baseline and a TripletFormer transformer. Key contributions include the first application of score-based diffusion to astronomical time series, a calibrated uncertainty–aware transformer alternative, and a realistic simulation suite that captures cross-band variability and lags. The framework yields high-fidelity reconstructions with quantified uncertainties, enabling robust cross-band lag analysis and physical inferences about accretion and emission near the black hole. Formally, we seek to infer latent signals $X^{(i)}(t)$ from sparse observations ${Y^{(i)}(t_k)}$ with a posterior $p(X|Y)$ realized by diffusion priors.

Abstract

Understanding the variability of Sagittarius A* (Sgr A*) requires coordinated, multi-wavelength observations that span the electromagnetic spectrum. In this work, we focus on data from four key observatories: Chandra in the X-ray (2-8 keV), GRAVITY on the Very Large Telescope in the near-infrared (2.2 microns), Spitzer in the infrared (4.5 microns), and ALMA in the submillimeter (340 GHz). These multi-band observations are essential for probing the physics of accretion and emission near the black hole's event horizon, yet they suffer from irregular sampling, band-dependent noise, and substantial data gaps. These limitations complicate efforts to robustly identify flares and measure cross-band time lags, key diagnostics of the physical processes driving variability. To address this challenge, we introduce a diffusion-based generative model, for interpolating sparse, multivariate astrophysical time series. This represents the first application of score-based diffusion models to astronomical time series. We also present the first transformer-based model for light curve reconstruction that includes calibrated uncertainty estimates. The models are trained on simulated light curves constructed to match the statistical and observational characteristics of real Sgr A* data. These simulations capture correlated multi-band variability, realistic observation cadences, and wavelength-specific noise. We compare our models against a multi-output Gaussian Process. The diffusion model achieves superior accuracy and competitive calibration across both simulated and real datasets, demonstrating the promise of diffusion models for high-fidelity, uncertainty-aware reconstruction of multi-wavelength variability in Sgr A*.

Figures (5)

• Figure 1: Conceptual illustration of probabilistic interpolation as an inverse modeling task. Starting from sparse and noisy observations $\{\mathbf{y}^{(i)}(t_k)\}$ (left), the process (model of choice) reconstructs the underlying continuous signal $\mathbf{x}^{(i)}(t)$ (right). The indices $(i)$ refer to different light curves. The shaded areas highlight the quantified uncertainty, demonstrating how the model addresses ambiguity in unobserved data regions. Crucially, information is shared across light curves: observations in one band are used to inform the interpolation of others, and vice versa.
• Figure 2: Multi-wavelength observational data of Sgr A* from the July 17–18, 2019 campaign, adapted from boyce_multiwavelength_2022. The panels show flux measurements in four distinct bands: X-ray (Chandra, 2–8 keV, top), NIR (GRAVITY, 2.2 $\mu$m, second), IR (Spitzer, 4.5 $\mu$m, third), and submm (ALMA, 340 GHz, bottom). These data exhibit irregular sampling and varying noise levels across modalities, with dense temporal coverage in X-ray and IR bands and sparser, noisier measurements in NIR and submm.
• Figure 3: Example of simulated multi-wavelength light curves used for training and evaluation, drawn randomly from the test set. Each panel corresponds to one wavelength channel (X-ray, NIR, IR, submm), and displays a sample light curve generated to mimic the variability patterns, noise characteristics, and irregular sampling in real Sgr A* observations. The simulations are adapted from the framework described in witzel_rapid_2021. A masking procedure is added to the simulated light curve to replicate telescope observation cadences. Colored circles represent observed (unmasked) data points and black circles represent masked data points.
• Figure 4: Architecture of the TripletFormer model (yalavarthi_tripletformer_2023). The encoder processes observed inputs via attention to produce a latent representation $Z$. The decoder uses cross-attention with target queries to produce predictive means $\mu$ and standard deviations $\sigma$, defining a Gaussian distribution over missing values.
• Figure 5: Model interpolation results on the simulated light curve of Sgr A* from Figure \ref{['fig:simulationdata']}, plotted from 4:00 UTC to 14:00 UTC for clarity. Each column corresponds to a different model (MOGP, TripletFormer, CSPD), and each row represents a different wavelength band (X-ray, NIR, IR, submm). Observed (unmasked) data upon which the models are conditioned are shown as grey circles, while test (masked) data points are shown as black circles. Solid lines denote the mean predictions of each model while shaded regions denote 2$\sigma$ credible intervals. All models successfully recover the underlying signal, but the CSPD model captures sharper variability and exhibits tighter uncertainty bounds compared to the smoother, more conservative predictions of the MOGP and TripletFormer. See Figure \ref{['fig:simulated data interpolation']} for the full 24 hour interpolation.