Uncertainty quantification is essential for deploying machine-learning models in real-world settings, yet standard neural networks provide only point estimates without uncertainty. Bayesian neural networks (BNNs) address this limitation by modeling predictive distributions, but their adoption remains restricted by the high computational cost of variational inference and sampling-based methods. This project aims to advance the mathematical foundations of BNNs by developing improved loss formulations, efficient input-error propagation techniques, and non-sampling inference approaches to reduce computational complexity. The proposed methods will be applied to photovoltaic power-generation forecasting, enabling more reliable uncertainty estimates for energy scheduling and resource management in renewable-energy systems.