# Developing Distance-Aware Physics-Constrained Probabilistic Frameworks for Industrial Prognostics

arXiv: https://arxiv.org/abs/2512.08499 · HTML: https://mlsanity.com/paper/2512.08499/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 9 of 9)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [Reverse KL-Divergence Training of Prior Networks: Improved Uncertainty and Adversarial Robustness](https://mlsanity.com/paper/1905.13472/) (2019)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [HUST bearing: a practical dataset for ball bearing fault diagnosis](https://mlsanity.com/paper/2302.12533/) (2023)
- [CARLE: A Hybrid Deep-Shallow Learning Framework for Robust and Explainable RUL Estimation of Rolling Element Bearings](https://mlsanity.com/paper/2510.17846/) (2025)

## Built on by (most cited first, 1 of 1)

- [FLUID: Continuous-Time Hyperconnected Sparse Transformer for Sink-Free Learning](https://mlsanity.com/paper/2605.04421/) (2026)

## Abstract (arXiv)

Development of reliable and physically interpretable probabilistic frameworks for industrial prognostics remain nascent, and existing literature is often insensitive as inputs move away from the training manifold. 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. We also introduce a distance-aware-coefficient~(DAC) metric to quantify sensitivity to distributional shifts. Empirically, we validate both frameworks on rolling-element-bearings (REBs) prognostics using the PRONOSTIA, XJTU-SY, and HUST benchmark datasets. Experimental results demonstrate improved prediction accuracy and well-calibrated uncertainty estimates relative to competing baselines, while maintaining auditable performance in cross-validation and robustness under extreme adversarial perturbations.