Deep learning methods have recently demonstrated remarkable success in scientific inverse problems, yet most approaches yield a single best reconstruction without an explicit and robust representation of the associated uncertainty. This limitation is critical in high-stakes settings, where overconfident errors and limited generalization can have significant consequences. Bayesian inference offers a principled way to describe plausible solutions and their relative support given data and prior information, yet it remains computationally demanding in high-dimensional settings. This thesis investigates a range of deep learning–assisted Bayesian approaches that combine learned representations with physical forward models to obtain uncertainty-aware reconstruction methods with tractable computational complexity. The first part of the thesis introduces IntraSeismic, an implicit neural representation (INR) framework for deterministic seismic inversion. One or more subsurface property models are parameterized as a coordinate-based neural network, which acts as a nonlinear preconditioner for the inversion and imposes an implicit bias to the model representation. This approach is subsequently extended to the Bayesian setting and is named B-IntraSeismic. By combining INRs with Variational Inference, I derive two tractable posterior approximations that quantify the point-wise uncertainty of the reconstructed subsurface features. Next, the practical value of these uncertainty estimates is validated in a downstream application, namely reservoir history matching for CO2 storage. I show that probabilistic seismic inversion results can be effectively assimilated into fluid flow simulations to update geological facies models, thereby propagating observation uncertainty into dynamic forecasting.  The final part of this thesis is devoted to HSDiff, a diffusion-based deep prior model for hyperspectral imaging. An unconditional diffusion model trained on hyperspectral images is used alongside an RGB image and the modelling operator of the RGB formation process via diffusion posterior sampling to generate approximate posterior samples of hyperspectral cubes. This framework is used to study the effect of different spectral encoding operators on the posterior distribution and to support the experimental design of sensor configurations with calibrated uncertainty. Collectively, these contributions establish a methodological progression from implicit parameterization to generative sampling, demonstrating that deep learning provides robust, physics-constrained mechanisms for uncertainty quantification in scientific inverse problems.

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