Dynamic Black-hole Emission Tomography with Physics-informed Neural Fields

TL;DR

This work addresses the ill-posed problem of dynamic 4D tomography around black holes using sparse EHT measurements. It introduces PI-DEF, a physics-informed neural-field framework that jointly learns a time-dependent emissivity field $e(t,\mathbf{x})$ and a 3D velocity field $\tilde{\mathbf{u}}(\mathbf{x})$, with soft constraints drawn from adaptive velocity models. Compared to BH-NeRF and physics-agnostic baselines, PI-DEF yields significantly improved emissivity reconstructions and more accurate velocity in regions with moving gas, while also enabling prospects for inferring black-hole spin from data. The approach demonstrates robustness to velocity-model mismatch, accommodates new emission over time, and lays groundwork for applying dynamic, physics-informed reconstructions to real EHT data with realistic noise and atmospheric effects.

Abstract

With the success of static black-hole imaging, the next frontier is the dynamic and 3D imaging of black holes. Recovering the dynamic 3D gas near a black hole would reveal previously-unseen parts of the universe and inform new physics models. However, only sparse radio measurements from a single viewpoint are possible, making the dynamic 3D reconstruction problem significantly ill-posed. Previously, BH-NeRF addressed the ill-posed problem by assuming Keplerian dynamics of the gas, but this assumption breaks down near the black hole, where the strong gravitational pull of the black hole and increased electromagnetic activity complicate fluid dynamics. To overcome the restrictive assumptions of BH-NeRF, we propose PI-DEF, a physics-informed approach that uses differentiable neural rendering to fit a 4D (time + 3D) emissivity field given EHT measurements. Our approach jointly reconstructs the 3D velocity field with the 4D emissivity field and enforces the velocity as a soft constraint on the dynamics of the emissivity. In experiments on simulated data, we find significantly improved reconstruction accuracy over both BH-NeRF and a physics-agnostic approach. We demonstrate how our method may be used to estimate other physics parameters of the black hole, such as its spin.

Figures (10)

• Figure 1: Our approach solves a severely ill-posed tomography problem in astrophysics. Given sparse EHT telescope measurements (visualized here as images), we recover a 4D emissivity field of the moving gas near a black hole. Our approach imposes soft physics constraints to constrain the reconstruction and recovers a 3D velocity field that explains the observed motion.
• Figure 2: Main components overview. For every time $t$, we project the estimated emissivity $e(t,\mathbf{x})$ onto the image plane and simulate EHT measurements. A data-fit loss makes sure that the resulting measurements agree with the observed EHT measurements. We propagate $e(t,\mathbf{x})$ forward in time by $\Delta t$ via the velocity network and an ODE solver. A dynamics loss checks that the resulting $\hat{e}(t+\Delta t,\mathbf{x})$ agrees with the $e(t+\Delta t,\mathbf{x})$ given by the emissivity network. A velocity regularization loss supervises the velocity network with an assumed velocity model.
• Figure 3: Example reconstructions of two random simulated ground-truth emissivity fields. 4D-MLP is a physics-agnostic approach (i.e., the 4D MLP is just fit to the data). BH-NeRF levis2022gravitationally is the previous approach that enforces a strict Keplerian velocity prior, which is incorrect in this region near the black hole. Our physics-informed approach gives the most accurate reconstructions.
• Figure 4: The $(u,v)$-coverage of the three arrays considered in our experiments. EHT 2025 marginally improves upon EHT 2017, while ngEHT significantly improves coverage. When simulating measurements, we kept the same bandwidth (2 GHz) for all three arrays, although ngEHT should have greater bandwidth (16 GHz).
• Figure 5: Emissivity reconstructions as measurements become sparser. "Image" corresponds to directly fitting to "Obs. image."