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In this paper, we develop two sampling-free, distance-aware physics-constrained probabilistic frameworks: (i) PC-SNGP and (ii) PC-SNER. Both apply spectral normalization to hidden layer weights, enforcing bi-Lipschitz distance-preserving representation from the input to the latent space. PC-SNGP replaces the dense output with Gaussian process whose posterior variance increases with input distance from the training manifold. PC-SNER modifies the output layer to predict Normal-Inverse-Gamma~(NIG) parameters for distance preserving estimation. To maintain balance between data fidelity and physical consistency during training, we introduce a dynamic weighting strategy for the physics-constrained loss. 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PC-SNER modifies the output layer to predict Normal-Inverse-Gamma~(NIG) parameters for distance preserving estimation. To maintain balance between data fidelity and physical consistency during training, we introduce a dynamic weighting strategy for the physics-constrained loss. 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