## 2 Materials and Methods
### Regularization and Uncertainty Quantification Module
By integrating various network modules and optimization strategies, the model is capable of accurately quantifying and calibrating these uncertainties, which is crucial for addressing the challenges of complex image super-resolution tasks. Stage 1: Epistemic Uncertainty via MC-Dropout  
Within the Bayesian deep learning framework, we quantify epistemic uncertainty through a Monte Carlo (MC) approximation approach. By retaining dropout layer activation during inference and performing T = 50 stochastic forward passes, this random sampling process approximates the posterior distribution mathematically as  
\[\mu(x)=\frac{1}{T}\sum_{t=1}^{T}f_{\theta}(x^{(t)}) \tag{7}\]  
\[\sigma_{epistemic}^{2}(x)=\frac{1}{T-1}\sum_{t=1}^{T}\left(f_{\theta}(x^{(t)}) -\mu(x)\right)^{2} \tag{8}\]  
Here, \(x^{(t)}\) denotes the input with stochastic perturbation at the \(t\)-th sampling iteration, and \(f_{\theta}(x^{(t)})\) represents the network mapping function parameterized by \(\theta\). The mean estimate \(\mu(x)\) aggregates distribution characteristics across parameter space through multi-sample averaging, while the variance term \(\sigma_{epistemic}^{2}(x)\) quantifies parametric uncertainty. We employ unbiased sample variance estimation (using \(T\)\(-\)1 denominator) to ensure statistical validity under limited sampling. Spatial heterogeneity in \(\sigma_{epistemic}^{2}(x)\) becomes prominent along complex topographic boundaries like submarine cliffs and volcanic cones, reflecting prediction multiplicity. Stage 2: Aleatoric Uncertainty via Heteroscedastic Noise  
A dedicated noise prediction branch is introduced to estimate the data-dependent variance \(\sigma_{aletoric}^{2}\). The model is trained using a joint loss function defined as  
\[L_{NLL}=\frac{1}{2N}\sum_{i=1}^{N}[\frac{(y_{i}-\mu_{i})^{2}}{\sigma_{aletoric }^{2}}+log(\sigma_{aletoric}^{2})] \tag{9}\]  
In this case, \(\mu_{i}\) is the predicted mean, \(\sigma_{aletoric}^{2}\) is the uncertainty estimate, and \(y_{i}\) is the ground truth. Lightweight modules like MultiScaleVarianceFusion and VarianceEstimator (Figure 1g,h) compute and combine variance maps from multiple scales to refine aleatoric uncertainty estimates. The VarianceEstimator is a lightweight network designed to estimate variance, operating on multi-scale features to produce a single-channel uncertainty map that reflects the predicted variance for each pixel. Meanwhile, the MultiScaleVarianceFusion module utilizes PixelShuffle up-sampling to merge variance maps estimated at different scales, achieving precise quantification of aleatoric uncertainty. The final joint loss function integrates negative log-likelihood, structural similarity (SSIM), and epistemic uncertainty regularization terms. \(L_{NLL}\) balances reconstruction error and variance estimation to prevent overconfidence, which predicts the mean as \(\mu_{i}\), estimates the uncertainty (variance) as \(\sigma_{i}^{2}\), and compares it to the true value \(y_{i}\):  
\[L_{NLL}=\frac{1}{N}\sum_{i=1}^{N}\big{[}\frac{(y_{i}-\mu_{i})^{2}}{2\sigma_{i}^{ 2}}+\frac{1}{2}log\sigma_{i}^{2}\big{]} \tag{10}\]  
SSIM loss (\(L_{SSIM}\)) ensures structural consistency by comparing the predicted mean with the ground truth:  
\[L_{SSIM}=1-SSIM(y,\mu) \tag{11}\]  
where \(y\) and \(u\) are the true image and predicted mean, respectively. Additionally, an epistemic uncertainty regularization term is incorporated to further suppress overfitting and enhance robustness:  
\[L_{epistemic}=\gamma\cdot\frac{1}{N}\sum_{i=1}^{N}log(mc\_var_{i}+e) \tag{12}\]  
Here, \(\gamma\) is the regularization weight, \(mc\_var_{i}\) represents the epistemic uncertainty at each location, \(e\) is a small constant to prevent numerical instability, and \(N\) is the number of pixels or samples averaged.