Super-resolution for low-fidelity physical system observations with probabilistic wavelet representation

Accurately reconstructing high-resolution physical fields from coarse partial differential equation observations remains a central challenge in computational mechanics, especially for problems involving multiscale structures, discontinuities, and non-Gaussian uncertainty. This work proposes the Probabilistic Wavelet Representation Model (PWRM) for uncertainty-aware super-resolution. The proposed method uses wavelets to capture multiscale representations with heavy-tailed priors embedding, and ultimately employs Stein variational gradient descent posterior inference to enhance reconstruction fidelity and uncertainty calibration. By learning sparse wavelet coefficients within a probabilistic framework, the proposed method achieves high-fidelity reconstruction while delivering calibrated epistemic uncertainty. Extensive experiments on five representative benchmarks demonstrate consistent improvements of approximately 0.6-1.9 dB in peak signal-to-noise ratio over strong deterministic and probabilistic baselines, including Poisson, multiscale diffusion, Euler shocks, fractional Laplacian fields, and phase-field fracture. Moreover, the inferred posterior variance reliably identifies regions of physical complexity such as shocks, interfaces, and crack tips, providing interpretable and meaningful epistemic uncertainty information. These results highlight PWRM as a generalizable, physically grounded, and uncertainty-aware super-resolution framework that effectively bridges Bayesian learning with computational physics.