Theory of Fokker–Planck Equations for Reliability and Remaining Useful Lifetime Prognostics

A theory to predict the reliability evolution and remaining useful lifetime (RUL) of technical devices in the framework of Fokker–Planck equations (FPE) is discussed and illustrated with examples. The FPE is a partial differential equation for the probability density in the space of aging model variables. This stochastic physics approach to prognostics is based on the mathematical equivalence of the FPE and Langevin equations covering, e.g., Brownian motion (BM) models with drift. Given an aging model, the FPE is an efficient framework for simulating the time dependent reliability, which is obtained from integration of the probability density in the acceptance region in condition space. Wear-out failures are associated with the probability flux out of the acceptance region through boundaries defined by end-of-life conditions. Chance failures can be included with sink terms in the FPE. Additional information gain by, e.g., condition monitoring (CM), can be considered by updating the probability density via Bayesian inference. The concept is useful if the deterministic part of the aging model can be formulated in terms of ordinary differential equations for aging variables, which requires a certain understanding of the physics of degradation and failure. To illustrate the concept, simulation results with own or commercial (here COMSOL Multiphysics) solvers for partial differential equations are presented for simple examples, including BM models with drift, stress–strength reliability, and battery aging.