Industrial processes often generate complex, interdependent time-series data from multiple sensors across multiple stages, forming complex dependencies among variables and process stages. Effective monitoring and timely anomaly detection of these time series through multivariate time series anomaly detection (MTAD) is crucial for preventing failures and ensuring the reliability of automated systems. Graph neural networks (GNNs) have advanced MTAD by leveraging data-driven graphs to model complex dependencies among variables, effectively capturing relational structures within multivariate time series to enhance anomaly detection performance. However, existing GNN-based approaches often overlook critical process knowledge, and even when this knowledge is considered, seamlessly incorporating it into existing models remains inherently challenging, leading to suboptimal performance. To address this limitation, we propose a knowledge-assisted multi-graph framework for modeling sensor dependencies in multi-stage industrial processes for MTAD, which explicitly incorporates process knowledge into graph learning to enhance dependency modeling and improve anomaly detection performance. Our method constructs three complementary graphs: one purely data-driven and two refined by integrating structural constraints derived from process knowledge. To effectively leverage these graphs for anomaly detection, we employ a multi-graph attention network, enabling a more accurate and robust representation of complex dependencies. Comprehensive experiments on two real-world, multi-stage industrial datasets demonstrate that incorporating process knowledge substantially enhances anomaly detection performance. Note to Practitioners-This work tackles anomaly detection in multi-stage industrial processes, where many sensors exhibit dependencies both within and across stages. Methods that learn sensor relations purely from data often overlook critical dependencies or infer spurious ones. To address this limitation, the proposed approach incorporates two forms of commonly available process knowledge-sensor group knowledge (sensors associated with each sub-process) and process flow knowledge (adjacent sub-process relationships)-derived from process diagrams. The method builds three complementary graphs (uninformed, sensor group knowledge-informed, and process flow knowledge-informed) and uses a multi-graph attention network combined with a temporal convolutional network for forecasting-based anomaly detection. Experiments on water treatment and water distribution systems show substantial F1 improvements over data-driven baselines, ablation studies confirming the contributions of both knowledge sources. For deployment, practitioners only need to specify sensor groupings and stage adjacencies from existing diagrams; no manual relationship modeling or specialized domain expertise is required. The model is trained on normal-operation data and detects anomalies via prediction errors. The approach is most effective when sub-process boundaries and flows are well documented and should be updated if the plant configuration changes. Beyond water systems, this method can be applied to chemical processing, power generation, and automated manufacturing facilities with similar multi-stage structures.

Uncertainty quantification (UQ) is crucial for deploying machine learning models in high-stakes applications, where overconfident predictions can lead to serious consequences. An effective UQ method must balance computational efficiency with the ability to generalize across diverse scenarios. Evidential deep learning (EDL) achieves efficiency by modeling uncertainty through the prediction of a Dirichlet distribution over class probabilities. However, the restrictive assumption of Dirichlet-distributed class probabilities limits EDL's robustness, particularly in complex or unforeseen situations. To address this, we propose *flexible evidential deep learning* ($\mathcal{F}$-EDL), which extends EDL by predicting a flexible Dirichlet distribution—a generalization of the Dirichlet distribution—over class probabilities. This approach provides a more expressive and adaptive representation of uncertainty, significantly enhancing UQ generalization and reliability under challenging scenarios. We theoretically establish several advantages of $\mathcal{F}$-EDL and empirically demonstrate its state-of-the-art UQ performance across diverse evaluation settings, including classical, long-tailed, and noisy in-distribution scenarios.

Evidential deep learning (EDL) has shown remarkable success in uncertainty estimation. However, there is still room for improvement, particularly in out-of-distribution (OOD) detection and classification tasks. The limited OOD detection performance of EDL arises from its inability to reflect the distance between the testing example and training data when quantifying uncertainty, while its limited classification performance stems from its parameterization of the concentration parameters. To address these limitations, we propose a novel method called Density Aware Evidential Deep Learning (DAEDL). DAEDL integrates the feature space density of the testing example with the output of EDL during the prediction stage, while using a novel parameterization that resolves the issues in the conventional parameterization. We prove that DAEDL enjoys a number of favorable theoretical properties. DAEDL demonstrates state-of-the-art performance across diverse downstream tasks related to uncertainty estimation and classification.