Multiscale Adaptive Network (MANet)

• MANet is a deep-learning framework that estimates six black-hole parameters from GRRT images using a ResNet backbone with integrated ACA and MEFP modules.
• The architecture employs Adaptive Channel Attention to enhance features tied to photon rings and accretion flows while preserving fine-to-coarse spatial details.
• It demonstrates robust performance under noisy conditions and limited data, making it valuable for astrophysical image regression and parameter inference.

Multiscale Adaptive Network (MANet) is a deep-learning regression framework for estimating black-hole physical parameters directly from General Relativistic Ray Tracing (GRRT) images. It was introduced for high-precision inference in settings characterized by limited observational resolution, high observational cost, sparse parameter spaces, and noisy or scarce samples. MANet combines a ResNet-based hierarchical feature extractor with two task-specific components—Adaptive Channel Attention (ACA) and a Multiscale Enhancement Feature Pyramid (MEFP)—to emphasize physically informative regions such as photon rings and inner accretion flows while preserving fine-to-coarse spatial structure relevant to parameter recovery from Event Horizon Telescope (EHT)-like images (Wei et al., 21 Jul 2025).

1. Problem formulation and GRRT data regime

The model is defined for six-parameter regression from simulated black-hole images. The target variables are black hole spin $a$, black hole mass $M_{BH}$, electron temperature $T_e$, accretion disk thickness $h_{disk}$, Keplerian factor $k$, and position angle $\mathrm{PA}$. The regression space is 6-dimensional and, for dataset generation, is uniformly sampled over the specified physical ranges.

Parameter	Range
$a$	$[-1, 1]$
$M_{BH}$	$[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$
$M_{BH}$0	$M_{BH}$1
$M_{BH}$2	$M_{BH}$3
$M_{BH}$4	$M_{BH}$5
$M_{BH}$6	$M_{BH}$7

The GRRT dataset contains 2157 images, each paired with ground-truth $M_{BH}$8. Simulations emulate EHT-like observing conditions at 230 GHz. Each image has a $M_{BH}$9 microarcsecond field of view on a $T_e$0 pixel grid, with observer inclination fixed at $T_e$1. Total flux is normalized to $T_e$2 Jy per image, and disk rotation direction is randomized. The paper frames data scarcity, sparse parameter spaces, and imbalanced positive/negative samples as general challenges in EHT-like scenarios; however, the constructed GRRT dataset itself uses uniform sampling across the six parameters together with stratified random splitting into 80% training (1725), 10% validation (216), and 10% test (216), explicitly to maintain distributional consistency across sets (Wei et al., 21 Jul 2025).

Label preprocessing is performed by parameter-wise standardization. For parameter dimension $T_e$3 with samples $T_e$4,

$T_e$5

and each target is transformed as

$T_e$6

This normalization makes the six regression targets comparable during training. Original physical values can be recovered from predictions using the stored normalization statistics.

2. Network backbone and end-to-end feature flow

MANet is built on a ResNet-based hierarchical feature extractor, with ResNet50 used in the reported experiments. The backbone follows the residual-network formulation of He et al. (He et al., 2015). In the MANet instantiation, residual blocks progressively downsample while enriching channel capacity, producing intermediate feature maps $T_e$7 (Wei et al., 21 Jul 2025).

The architecture inserts two specialized modules into intermediate feature stages. ACA performs channel-wise modulation using global context so that channels responsive to physically informative structures can be selectively enhanced. MEFP then applies parallel convolutions with different receptive fields—$T_e$8, $T_e$9, and $h_{disk}$0—to preserve and fuse fine-to-coarse structures such as photon ring edges, inner-flow asymmetries, and larger accretion-disk morphology. A learnable channel-wise gate merges the enhanced multiscale representation with the original residual feature.

The forward path is described at diagram level as follows: the $h_{disk}$1 image is first processed by early ResNet blocks to produce an intermediate feature tensor $h_{disk}$2; ACA computes global channel descriptors and reweights $h_{disk}$3 to obtain $h_{disk}$4; MEFP transforms $h_{disk}$5 through three parallel branches and fuses them into $h_{disk}$6; and a channel-wise sigmoid gate blends $h_{disk}$7 with the original $h_{disk}$8 to form $h_{disk}$9, which continues through the backbone and finally reaches the regression head for parameter estimation. Exact module insertion points within ResNet50 are not specified.

This organization makes MANet a feature-adaptive regression model rather than a purely generic CNN regressor. A plausible implication is that the design aims to reduce information loss from downsampling while retaining sensitivity to compact image structures that encode spacetime and accretion-flow properties.

3. Adaptive Channel Attention (ACA)

ACA is motivated by the observation that GRRT images contain brightness asymmetries, thin photon rings, and jet/disk boundaries, not all of which are equally informative for the target parameters. The module computes a global descriptor for each channel and uses it to modulate the feature tensor channel-wise (Wei et al., 21 Jul 2025).

Given $k$0, ACA first performs spatial averaging:

$k$1

The descriptor is then passed through a fully connected layer:

$k$2

The attention-enhanced feature is obtained by broadcasted channel-wise scaling with a learnable factor $k$3:

$k$4

where $k$5 is reshaped to $k$6 before elementwise multiplication with $k$7. The paper does not specify additional nonlinearities in ACA beyond the learnable linear transform $k$8.

Within MANet, ACA precedes multiscale processing. Its stated role is to strengthen channels that respond to physically diagnostic structures, including photon-ring signatures, beaming asymmetries, and orientation cues. Grad-CAM visualizations reported in the study show that ACA concentrates activations on thin photon-ring segments and high-emission inner-disk regions, producing sharper and more spatially selective heatmaps than the baseline. This supports the interpretation that ACA improves alignment between learned features and the image regions most relevant to spacetime geometry and accretion physics.

4. Multiscale Enhancement Feature Pyramid (MEFP) and gated fusion

MEFP addresses the multiscale nature of GRRT imagery. The paper explicitly motivates it by the coexistence of thin, high-contrast photon-ring edges, mid-scale brightness asymmetries and disk thickness cues, and large-scale accretion-disk morphology. Single-kernel processing is described as insufficient because downsampling can suppress precisely those structures needed for accurate regression (Wei et al., 21 Jul 2025).

The module takes $k$9 as input and processes it through three parallel branches:

$\mathrm{PA}$0

$\mathrm{PA}$1

$\mathrm{PA}$2

Here $\mathrm{PA}$3 denotes ReLU in MEFP. The $\mathrm{PA}$4 and $\mathrm{PA}$5 branches use zero padding to preserve spatial dimensions for later fusion. The three branch outputs are concatenated across channels,

$\mathrm{PA}$6

and reduced back to the original channel count with a $\mathrm{PA}$7 fusion layer:

$\mathrm{PA}$8

The per-branch channel count $\mathrm{PA}$9 is not specified numerically.

Fusion with the original residual feature is controlled by a learnable channel-wise gate:

$a$0

where $a$1 and $a$2 is sigmoid applied channel-wise. This construction explicitly balances enhanced multiscale features against the unmodified backbone representation. The paper’s module-level pseudocode uses the same sequence: global pooling, fully connected attention, multibranch convolutions, concatenation, $a$3 fusion, and gated residual blending.

5. Optimization protocol and regression performance

The reported training protocol compares four learning-rate schedules under a unified setup: constant, step, exponential, and cosine annealing. All are trained for 500 epochs with an initial learning rate of $a$4. Cosine annealing achieves the best validation $a$5 across all six parameters and is adopted for the main experiments. Optimizer type, batch size, data augmentation, initialization, normalization layers, hardware, and the exact training loss function are not specified. The paper also does not mention early stopping; model selection is discussed through validation $a$6 comparisons (Wei et al., 21 Jul 2025).

Evaluation uses the coefficient of determination,

$a$7

where $a$8 are ground-truth values, $a$9 are predictions, and $[-1, 1]$0. The metric is reported per parameter. For the learning-rate comparison, cosine annealing yields validation $[-1, 1]$1 values of $[-1, 1]$2 for $[-1, 1]$3, $[-1, 1]$4 for $[-1, 1]$5, $[-1, 1]$6 for $[-1, 1]$7, $[-1, 1]$8 for $[-1, 1]$9, $M_{BH}$0 for $M_{BH}$1, and $M_{BH}$2 for $M_{BH}$3. For spin $M_{BH}$4, the final reported losses under the four schedules are $M_{BH}$5 (constant), $M_{BH}$6 (step), $M_{BH}$7 (exponential), and $M_{BH}$8 (cosine), although the loss itself is not defined.

Ablation and baseline results on the simulated dataset show systematic gains from the specialized modules:

• ResNet50: $M_{BH}$9 0.9359, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$0 0.9284, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$1 0.9708, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$2 0.9986, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$3 0.9556, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$4 0.8569.
• ResNet50+MEFP: $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$5 0.9887, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$6 0.9819, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$7 0.9871, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$8 0.9992, $[2\times 10^9 M_\odot,\ 10^{10} M_\odot]$9 0.9603, $M_{BH}$00 0.9181.
• ResNet50+ACA: $M_{BH}$01 0.9864, $M_{BH}$02 0.9759, $M_{BH}$03 0.9912, $M_{BH}$04 0.9990, $M_{BH}$05 0.9561, $M_{BH}$06 0.9459.
• MANet (ACA+MEFP): $M_{BH}$07 0.9906, $M_{BH}$08 0.9969, $M_{BH}$09 0.9927, $M_{BH}$10 0.9993, $M_{BH}$11 0.9725, $M_{BH}$12 0.9602.

The paper attributes different improvements to the two modules. MEFP significantly boosts parameters driven by multiscale spatial gradients, including $M_{BH}$13 by $M_{BH}$14 over the ResNet50 baseline and $M_{BH}$15 by $M_{BH}$16. ACA particularly improves parameters linked to directional or orientation cues, with $M_{BH}$17 improving by $M_{BH}$18 over baseline. Their combination gives the strongest overall performance, indicating synergy between multiscale spatial fusion and attentive channel emphasis.

6. Noise robustness, interpretability, and astrophysical relevance

Robustness is evaluated by injecting Gaussian noise at test time with zero mean and standard deviations of 10 and 20 $M_{BH}$19as-equivalent units, denoted Gauss10 and Gauss20. Under Gauss10, the baseline ResNet50 versus MANet results are: $M_{BH}$20 0.9444 $M_{BH}$21 0.9641, $M_{BH}$22 0.8876 $M_{BH}$23 0.9976, $M_{BH}$24 0.9538 $M_{BH}$25 0.9775, $M_{BH}$26 0.9927 $M_{BH}$27 0.9960, $M_{BH}$28 0.9754 $M_{BH}$29 0.9807, and $M_{BH}$30 0.8376 $M_{BH}$31 0.8998. Under Gauss20, the corresponding values are: $M_{BH}$32 0.6868 $M_{BH}$33 0.7029, $M_{BH}$34 0.8862 $M_{BH}$35 0.9862, $M_{BH}$36 $M_{BH}$37, $M_{BH}$38 0.9742 $M_{BH}$39 0.9744, $M_{BH}$40 0.9491 $M_{BH}$41 0.9580, and $M_{BH}$42 0.8507 $M_{BH}$43 0.8601 (Wei et al., 21 Jul 2025).

Several implications are stated directly. MANet consistently outperforms the baseline under noise, especially for $M_{BH}$44 and $M_{BH}$45. The parameter $M_{BH}$46 is the most noise-sensitive; its fine-grained morphological cues degrade rapidly as noise increases, driving $M_{BH}$47 to near zero at Gauss20. The combination of ACA and MEFP helps maintain performance under moderate perturbations, suggesting improved generalization when training data are limited and observations are noisy.

The reported interpretability results are centered on Grad-CAM. ACA causes heatmaps to focus more sharply on thin photon-ring segments and high-emission inner-disk regions than the baseline, supporting the claim that the network attends to physically informative structures rather than diffuse image content. This is relevant to EHT-oriented analysis because the model is intended for parameter inference from very long baseline interferometry reconstructions and GRRT models, where sparse and noisy imaging is the norm.

The paper also extends the scope beyond black-hole imaging. It proposes that the same multiscale attention design is applicable to other astrophysical imaging tasks characterized by sparse/noisy data and mixed fine-to-coarse structures, including jets, disks, and gravitational lensing features. This suggests a broader methodological role for MANet-like modules in scientific image regression.

7. Limitations, reproducibility, and open directions

The study identifies several limitations. It models six parameters only; additional physical parameters and temporal variability are not included. Uncertainty quantification is absent, and the paper states that future work will incorporate temporal features and uncertainty estimation. The sensitivity of $M_{BH}$48 under severe noise remains a specific unresolved issue, indicating that stabilizing fine-structure estimation is still necessary (Wei et al., 21 Jul 2025).

Reproducibility is partial rather than complete. The backbone is specified as ResNet50, the input size is $M_{BH}$49, labels are standardized by z-scoring, the split is stratified 80/10/10, and the preferred schedule is cosine annealing with initial learning rate $M_{BH}$50 for 500 epochs. The paper also states that no explicit augmentation is reported during training and that noise is injected only during robustness evaluation on the test set. However, the exact training objective, optimizer, batch size, normalization layers, initialization scheme, hardware, intermediate feature sizes, exact module insertion points, and computational metrics such as parameter count, FLOPs, inference time, and memory footprint are not reported.

Two clarifications are important for interpretation. First, the paper discusses imbalanced positive/negative samples as a challenge typical of observational scenarios, but the GRRT dataset used for the reported experiments is uniformly sampled over the six-parameter space and stratified across train/validation/test splits. Second, the qualitative argument that deep learning is more efficient than dense sampling in physics-based forward modeling is not accompanied by exact computational benchmarks. Accordingly, MANet should be understood as a learned regression framework validated on a controlled GRRT dataset under fixed observer inclination and normalized flux, rather than as a fully characterized deployment protocol for real EHT data.

Data availability is stated via Science Data Bank, with the DOI given in the paper, whereas code availability is not stated. Real EHT observations may exhibit noise characteristics and calibration systematics different from those assumed in the simulations. As a result, the reported results establish MANet’s behavior under a specific simulation regime and define a basis for subsequent work on temporal modeling, uncertainty estimation, and adaptation to observational systematics.