Mathematical Modeling of Industrial Prognosis and Health Management

Dear Colleagues,

Industrial prognostics and health management (PHM) are foundational to reliability, safety, and sustainability in modern cyber–physical industries, where fleets of assets produce massive, heterogeneous, and nonstationary data streams. This landscape raises core mathematical questions—how to model multi-scale degradation dynamics, reason under rare-failure and censored data, quantify uncertainty, and optimize maintenance policies under constraints—while coping with distribution shift, partial observability, and closed-loop interactions with control. Mathematics provides the unifying language for these challenges via stochastic and dynamical systems, state-space and PDE-constrained inference, Bayesian and causal reasoning, robust and distributionally robust optimization, graph signal processing, and sequential decision-making with guarantees. At the same time, rapid advances in artificial intelligence and deep learning offer complementary tools that can be made rigorous: physics-informed and operator-learning networks for gray-box modeling; probabilistic deep models with calibrated uncertainty; representation learning and time-series foundation models for degradation trajectories; interpretable and causal machine learning; domain adaptation and generalization across operating conditions; and safe reinforcement learning for maintenance scheduling with verifiable performance. This Special Issue seeks contributions that advance the mathematical foundations and computational methods of industrial PHM, and explicitly welcomes mathematically grounded AI and deep learning approaches that bridge theory and deployable practice.

This Special Issue aims to advance mathematical foundations and computational methods for industrial prognostics and health management (PHM). We invite contributions on the following:

1. Provable models for degradation, fault evolution, and remaining useful life—spanning stochastic processes, dynamical systems, and state-space and PDE-constrained inference;
2. Statistical inference under uncertainty and distribution shift, robust and distributionally robust optimization and control, and decision-theoretic formulations;
3. Mathematically grounded AI and deep learning—interpretable and causal ML, physics-/operator-informed networks, time-series foundation models, calibrated uncertainty, domain adaptation, and safe RL for maintenance;
4. Efficient and reproducible computation—numerical analysis, algorithmic complexity, parallel/graph computation, simulation, and benchmarking.

I look forward to receiving your contributions.

Dr. Lei Song
Guest Editor

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• condition monitoring and fault diagnosis
• prognostics and health management
• remaining useful life (RUL) estimation
• intelligent maintenance decision
• stochastic process modeling
• physics-informed neural network
• time series foundation model
• reinforcement learning
• transfer learning and domain generalization