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

Conference: CVPR2026
arXiv: 2602.08029
Code: Not open-sourced
Area: 3D Vision / Computational Imaging / Scientific Imaging
Keywords: Black hole imaging, Neural Radiance Fields, Physics-informed constraints, 4D Tomography, Event Horizon Telescope

TL;DR¶

Ours proposes PI-DEF, utilizing physics-informed coordinate neural networks to simultaneously reconstruct the 4D (time + 3D) emissivity field and 3D velocity field of gas near a black hole. Under sparse EHT measurements, it significantly outperforms BH-NeRF, which relies on hard-constrained Keplerian dynamics.

Background & Motivation¶

Static imaging is successful; dynamic 3D imaging is the next frontier: EHT has successfully captured static 2D images of M87 and Sgr A, but static images are complex 2D projections of 3D emissivity and cannot reveal the physical nature of the dynamic 3D environment.
Extremely underdetermined inverse problem: EHT can only observe from a single viewpoint, and measurements are highly sparse and corrupted by noise. 4D tomographic reconstruction is a severely ill-posed problem.
Dynamic sources increase difficulty: Radiating gas moves, appears, and disappears over time, making it impossible to simply aggregate measurements across time to improve reconstruction quality.
Forward model partially unknown: Light propagation depends on the unknown fluid dynamics near the black hole, and the measurement forward model is not fully known.
BH-NeRF assumptions are too strong: The previous sole method, BH-NeRF, assumes Keplerian dynamics. However, near the black hole, strong gravity and electromagnetic activity cause fluid dynamics to deviate from Keplerian models, and it cannot handle newly appearing radiation.
Significant scientific significance: Recovering the dynamic 3D emissivity field near a black hole can reveal unseen parts of the universe, helping to test General Relativity and infer physical parameters such as black hole spin.

Method¶

Overall Architecture¶

PI-DEF addresses an extremely ill-posed inverse problem: the Event Horizon Telescope (EHT) provides only single-view, sparse, and noisy observations of radiating gas near a black hole. The goal is to reconstruct its time-varying 4D (time + 3D) emissivity field along with a 3D velocity field. Previous work, BH-NeRF, assumed gas strictly followed Keplerian dynamics, which is untenable in strong gravity regions and fails to handle radiation appearing within the observation window.

PI-DEF uses two coordinate neural networks to represent the 4D emissivity field \(e(t, \mathbf{x}; \theta_e)\) and the 3D velocity field \(\tilde{u}^i(\mathbf{x}; \theta_v)\), respectively. Both employ positional encoding \(\gamma\) to enhance high-frequency expressiveness. These two fields are jointly optimized via three loss terms to fit observations while satisfying fluid continuity and soft physical priors. The total loss is \(\mathcal{L} = \lambda_{\text{data}}\mathcal{L}_{\text{data}} + \lambda_{\text{dyn}}\mathcal{L}_{\text{dyn}} + \lambda_{\text{reg}}\mathcal{L}_{\text{reg}}\). The key insight is that the physical model acts as a decayable soft regularization rather than a hard constraint.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    IN["EHT Sparse Noisy Observations<br/>Single-view visibility"]
    EN["Emissivity Field Network<br/>e(t,x;θe) + Positional Encoding γ"]
    VN["Velocity Field Network<br/>ũ(x;θv) + Positional Encoding γ"]
    PARAM["Velocity Parameterization<br/>Normal observer frame to avoid singularities"]
    VN --> PARAM
    IN --> LDATA["Data Fitting Loss L_data<br/>GRRT Projection<br/>Redshift g² derived from velocity"]
    EN --> LDATA
    PARAM --> LDATA
    EN --> LDYN["Dynamics Loss L_dyn<br/>ODE propagation e(t)→ê(t+Δt) via velocity<br/>L1 with e(t+Δt)"]
    PARAM --> LDYN
    PARAM --> LREG["Velocity Soft Regularization L_reg<br/>AART Prior · Exponentially decaying weight"]
    LDATA --> OPT["Joint Optimization<br/>L = λ_data·L_data + λ_dyn·L_dyn + λ_reg·L_reg"]
    LDYN --> OPT
    LREG --> OPT
    OPT --> OUT["4D Emissivity Field + 3D Velocity Field"]

Key Designs¶

1. Data Fitting Loss: Anchoring Two Fields to Observations via GRRT

This term aligns the reconstruction with real measurements. It projects the emissivity field onto the image plane using General Relativistic Ray Tracing (GRRT) to simulate EHT visibility measurements, performing a Gaussian likelihood fit with observations. The ingenuity lies in deriving the redshift factor \(g^2\) in the projection from the velocity field. Thus, a single \(\mathcal{L}_{\text{data}}\) constrains both the emissivity and velocity networks simultaneously—if the velocity estimation is wrong, the redshift is wrong, resulting in a poor fit. Geodesics are computed using the kgeo library, and measurements are simulated with eht-imaging.

2. Dynamics Loss: Linking Emissivity Across Time via Velocity Fields

This is the core innovation addressing the challenge that dynamic sources prevent simple temporal aggregation of measurements. It propagates the emissivity field \(e(t)\) over \(\Delta t\) using an ODE solver based on the predicted velocity from the velocity network to obtain \(\hat{e}(t+\Delta t)\). This is compared against the direct prediction \(e(t+\Delta t)\) via an L1 loss, forcing temporal self-consistency—how emissivity flows must match the velocity field. To prevent a "blurry blob" from bypassing this loss, Gaussian kernel blurring is applied before comparison. It essentially formulates fluid continuity as a differentiable consistency constraint.

3. Exponentially Decaying Velocity Soft Regularization: Trusting Physics Then Data

Under extreme underdetermination, velocity can rarely be recovered purely from data, yet hard-coding theoretical models repeats the failures of BH-NeRF. PI-DEF takes a middle ground: using the AART velocity model (including sub-Keplerian velocity and radial fall-in, controlled by \((\beta_\phi, \beta_r, \xi)\)) as a soft prior. An L2 regularization \(\mathcal{L}_{\text{reg}}\) is applied to the estimated velocity, but the weight decays exponentially during training (\(\lambda_{\text{init}}=10^6 \to \lambda_{\text{final}}=10\)). Early stages use the prior to guide velocity into a reasonable range, while later stages allow data to drive the refinement, enabling correction even if the prior doesn't perfectly match the true dynamics.

4. Numerically Stable Velocity Parameterization: Avoiding Singularities Near the Event Horizon

If velocity is directly estimated in the four-velocity frame, numerical issues arise where \(u^t\) is undefined. PI-DEF instead estimates in the numerically stable normal observer frame. Furthermore, the emissivity field is explicitly time-dependent, allowing it to capture radiation appearing within the observation window—a capability lacking in BH-NeRF, which only models an initial field with fixed propagation. Ablations show that velocity networks parameterized by radius \(r\), \((r, \theta)\), or \((x, y, z)\) all yield reasonable results, suggesting that axi-symmetric constraints are not strictly necessary.

Key Experimental Results¶

Emissivity Reconstruction Accuracy (5 random test scenarios, ngEHT measurements)¶

Method	PSNR (dB) ↑	MSE (×10⁻⁵) ↓
PI-DEF (Ours)	37.3 ± 2.3	2.3 ± 0.2
4D-MLP	35.4 ± 0.5	3.8 ± 0.4
BH-NeRF	34.0 ± 1.9	4.9 ± 0.8

Even when the velocity prior is mismatched (assuming pure sub-Keplerian without radial fall-in), PI-DEF significantly outperforms BH-NeRF and the purely data-driven 4D-MLP.
BH-NeRF fails severely near the black hole due to hard Keplerian constraints.

Ablation Study¶

Measurement Sparsity: ngEHT (23 telescopes) significantly improves reconstruction quality compared to EHT 2025 (12 telescopes) and EHT 2017 (8 telescopes). EHT 2025 offers limited improvement over 2017.
Velocity Recovery: In high emissivity density regions (>65th percentile), PI-DEF's radial and azimuthal velocity recovery matches the ground truth well, even with mismatched initial assumptions. Velocity recovery in low emissivity regions remains unconstrained.
Velocity Network Parameterization: Reasonable results are obtained using \(r\), \((r, \theta)\), or \((x, y, z)\) parameterizations, indicating axi-symmetric constraints are not essential.
Real Noise: Works under realistic Gaussian noise at Sgr A* total flux levels (~2.3 Jy).
Atmospheric Noise: Using closure phase and amplitudes instead of complex visibility handles atmospheric phase errors, though reconstruction accuracy decreases.
Spin Inference: The data fitting loss is sensitive to assumed black hole spin; the loss is minimized at the correct spin \(a=0.2\), proving PI-DEF can be used to infer physical parameters.

Highlights¶

Soft vs. Hard Constraints: Using physical velocity models as exponentially decaying soft regularization rather than hard constraints balances prior guidance with robustness against modeling errors. This design is elegant and generalizable.
Dual-field Joint Reconstruction: Simultaneously recovering the 4D emissivity field and 3D velocity field, with a dynamics loss establishing physical consistency between them.
Outstanding Scientific Value: Achievement of dynamic 3D reconstruction in regions very close to the black hole (not just distant flares) for the first time, demonstrating the potential for spin inference.
CV-driven Fundamental Physics: Combining the NeRF paradigm with GRRT geodesics serves as an exemplary case of CV technology pushing the frontiers of astrophysics.

Limitations & Future Work¶

Validated only on simulated data; not yet applied to real EHT data.
Ignores the slow-light effect (finite speed of light propagation) for gases moving at relativistic speeds.
Ignores attenuation due to absorption and scattering.
Velocity recovery is difficult very close to the event horizon where emissivity contribution is minimal due to the redshift factor \(g^2 \to 0\).
Currently does not jointly optimize black hole spin and inclination; only a grid-search proof-of-concept for spin was performed.
Gaussian Splatting was excluded due to its inability to handle the appearance and disappearance of hotspots, though its efficiency might be leveraged via dynamic point management in the future.

Method	Temporal Modeling	Velocity Constraint	New Emitters	Representative Work
BH-NeRF	Initial field + Keplerian propagation	Hard (Keplerian)	✗	ECCV 2022
4D-MLP	4D coordinate network	None	✓	Baseline
PI-DEF	4D coordinate network	Soft (AART decay reg.)	✓	Ours

Difference from standard NeRF dynamic scene methods: PI-DEF handles an extreme setting of curved light paths (GRRT) + single view + dynamic sources, rather than linear ray tracing + multi-view standard scenes.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ — First to introduce physics-informed soft constraints to black hole 4D tomography; dual-field joint reconstruction and decay regularization are novel.
Experimental Thoroughness: ⭐⭐⭐⭐ — Comprehensive simulations (sparsity, noise, spin inference, ablation), but lacks real-world data validation.
Writing Quality: ⭐⭐⭐⭐⭐ — Clear explanation of physical background and methods; high-quality visualizations suitable for non-astrophysics readers.
Value: ⭐⭐⭐⭐⭐ — A benchmark work of CV technology aiding fundamental physics with direct application prospects for EHT science.