Mathematical Framework for Digital Risk Twins in Safety-Critical Systems

This paper introduces a formal mathematical framework for Digital Risk Twins (DRTs) as an extension of traditional digital twin (DT) architectures, explicitly tailored to the needs of safety-critical systems. While conventional DTs enable real-time monitoring and simulation of physical assets, they often lack structured mechanisms to model stochastic failure processes; evaluate dynamic risk; or support resilient, risk-aware decision-making. The proposed DRT framework addresses these limitations by embedding probabilistic hazard modeling, reliability theory, and coherent risk measures into a modular and mathematically interpretable structure. The DT to DRT transformation is formalized as a composition of operators that project system trajectories onto risk-relevant features, compute failure intensities, and evaluate risk metrics under uncertainty. The framework supports layered integration of simulation, feature extraction, hazard dynamics, and decision-oriented evaluation, providing traceability, scalability, and explainability. Its utility is demonstrated through a case study involving an aircraft brake system, showcasing early warning detection, inspection schedule optimization, and visual risk interpretation. The results confirm that the DRT enables modular, explainable, and domain-agnostic integration of reliability logic into digital twin systems, enhancing their value in safety-critical applications.