Developing Distance-Aware, and Evident Uncertainty Quantification in Dynamic Physics-Constrained Neural Networks for Robust Bearing Degradation Estimation

Accurate and uncertainty-aware degradation estimation is essential for predictive maintenance in safety-critical systems like rotating machinery with rolling-element bearings. Many existing uncertainty methods lack confidence calibration, are costly to run, are not distance-aware, and fail to generalize under out-of-distribution data. We introduce two distance-aware uncertainty methods for deterministic physics-guided neural networks: PG-SNGP, based on Spectral Normalization Gaussian Process, and PG-SNER, based on Deep Evidential Regression. We apply spectral normalization to the hidden layers so the network preserves distances from input to latent space. PG-SNGP replaces the final dense layer with a Gaussian Process layer for distance-sensitive uncertainty, while PG-SNER outputs Normal Inverse Gamma parameters to model uncertainty in a coherent probabilistic form. We assess performance using standard accuracy metrics and a new distance-aware metric based on the Pearson Correlation Coefficient, which measures how well predicted uncertainty tracks the distance between test and training samples. We also design a dynamic weighting scheme in the loss to balance data fidelity and physical consistency. We test our methods on rolling-element bearing degradation using the PRONOSTIA, XJTU-SY and HUST datasets and compare them with Monte Carlo and Deep Ensemble PGNNs. Results show that PG-SNGP and PG-SNER improve prediction accuracy, generalize reliably under OOD conditions, and remain robust to adversarial attacks and noise.

Neuronal Attention Circuit (NAC) for Representation Learning

Attention improves representation learning over RNNs, but its discrete nature limits continuous-time (CT) modeling. We introduce Neuronal Attention Circuit (NAC), a novel, biologically plausible CT-Attention mechanism that reformulates attention logits computation as the solution to a linear first-order ODE with nonlinear interlinked gates derived from repurposing \textit{C. elegans} Neuronal Circuit Policies (NCPs) wiring mechanism. NAC replaces dense projections with sparse sensory gates for key-query projections and a sparse backbone network with two heads for computing \textit{content-target} and \textit{learnable time-constant} gates, enabling efficient adaptive dynamics. NAC supports three attention logit computation modes: (i) explicit Euler integration, (ii) exact closed-form solution, and (iii) steady-state approximation. To improve memory intensity, we implemented a sparse Top-\emph{K} pairwise concatenation scheme that selectively curates key-query interactions. We provide rigorous theoretical guarantees, including state stability, bounded approximation errors, and universal approximation. Empirically, we implemented NAC in diverse domains, including irregular time-series classification, lane-keeping for autonomous vehicles, and industrial prognostics. We observed that NAC matches or outperforms competing baselines in accuracy and occupies an intermediate position in runtime and memory efficiency compared with several CT baselines.