Quantification of Model-Form Uncertainty in Drift-Diffusion Simulation Using Fractional Derivatives

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest such as rare events difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker-Planck equation (fFPE) is proposed to describe the drift-diffusion processes under long-range correlations and memory effects. A new model calibration approach based on the maximum accumulative mutual information is also proposed to reduce model-form uncertainty, where an optimization procedure is taken.