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.

IBN: An Interpretable Bidirectional-Modeling Network for Multivariate Time Series Forecasting with Variable Missing

Multivariate time series forecasting (MTSF) often faces challenges from missing variables, which hinder conventional spatial-temporal graph neural networks in modeling inter-variable correlations. While GinAR addresses variable missing using attention-based imputation and adaptive graph learning for the first time, it lacks interpretability and fails to capture more latent temporal patterns due to its simple recursive units (RUs). To overcome these limitations, we propose the Interpretable Bidirectional-modeling Network (IBN), integrating Uncertainty-Aware Interpolation (UAI) and Gaussian kernel-based Graph Convolution (GGCN). IBN estimates the uncertainty of reconstructed values using MC Dropout and applies an uncertainty-weighted strategy to mitigate high-risk reconstructions. GGCN explicitly models spatial correlations among variables, while a bidirectional RU enhances temporal dependency modeling. Extensive experiments show that IBN achieves state-of-the-art forecasting performance under various missing-rate scenarios, providing a more reliable and interpretable framework for MTSF with missing variables. Code is available at: https://github.com/zhangth1211/NICLab-IBN.

C3RL: Rethinking the Combination of Channel-independence and Channel-mixing from Representation Learning

Multivariate time series forecasting has drawn increasing attention due to its practical importance. Existing approaches typically adopt either channel-mixing (CM) or channel-independence (CI) strategies. CM strategy can capture inter-variable dependencies but fails to discern variable-specific temporal patterns. CI strategy improves this aspect but fails to fully exploit cross-variable dependencies like CM. Hybrid strategies based on feature fusion offer limited generalization and interpretability. To address these issues, we propose C3RL, a novel representation learning framework that jointly models both CM and CI strategies. Motivated by contrastive learning in computer vision, C3RL treats the inputs of the two strategies as transposed views and builds a siamese network architecture: one strategy serves as the backbone, while the other complements it. By jointly optimizing contrastive and prediction losses with adaptive weighting, C3RL balances representation and forecasting performance. Extensive experiments on seven models show that C3RL boosts the best-case performance rate to 81.4\% for models based on CI strategy and to 76.3\% for models based on CM strategy, demonstrating strong generalization and effectiveness. The code will be available once the paper is accepted.

FMamba: Mamba based on Fast-attention for Multivariate Time-series Forecasting

In multivariate time-series forecasting (MTSF), extracting the temporal correlations of the input sequences is crucial. While popular Transformer-based predictive models can perform well, their quadratic computational complexity results in inefficiency and high overhead. The recently emerged Mamba, a selective state space model, has shown promising results in many fields due to its strong temporal feature extraction capabilities and linear computational complexity. However, due to the unilateral nature of Mamba, channel-independent predictive models based on Mamba cannot attend to the relationships among all variables in the manner of Transformer-based models. To address this issue, we combine fast-attention with Mamba to introduce a novel framework named FMamba for MTSF. Technically, we first extract the temporal features of the input variables through an embedding layer, then compute the dependencies among input variables via the fast-attention module. Subsequently, we use Mamba to selectively deal with the input features and further extract the temporal dependencies of the variables through the multi-layer perceptron block (MLP-block). Finally, FMamba obtains the predictive results through the projector, a linear layer. Experimental results on eight public datasets demonstrate that FMamba can achieve state-of-the-art performance while maintaining low computational overhead.