# On the representation of model inadequacy : a stochastic operator approach

## Abstract

Mathematical models of physical systems are subject to many sources of uncertainty such as measurement errors and uncertain initial and boundary conditions. After accounting for these uncertainties, it is often revealed that there remains some discrepancy between the model output and the observations; if so, the model is said to be inadequate. In practice, the inadequate model may be the best that is available or tractable, and so despite its inadequacy the model may be used to make predictions of unobserved quantities. In this case, a representation of the inadequacy is necessary, so the impact of the observed discrepancy can be determined. We investigate this problem in the context of chemical kinetics and propose a new technique to account for model inadequacy that is both probabilistic and physically meaningful. Chemical reactions are generally modeled by a set of nonlinear ordinary differential equations (ODEs) for the concentrations of the species and temperature. In this work, a stochastic inadequacy operator S is introduced which includes three parts. The first is represented by a random matrix which is embedded within the ODEs of the concentrations. The matrix is required to satisfy several physical constraints, and its most general form exhibits some useful properties, such as having only non-positive eigenvalues. The second is a smaller but specific set of nonlinear terms that also modifies the species’ concentrations, and the third is an operator that properly accounts for changes to the energy equation due to the previous changes. The entries of S are governed by probability distributions, which in turn are characterized by a set of hyperparameters. The model parameters and hyperparameters are calibrated using high-dimensional hierarchical Bayesian inference, with data from a range of initial conditions. This allows the use of the inadequacy operator on a wide range of scenarios, rather than correcting any particular realization of the model with a corresponding data set. We apply the method to typical problems in chemical kinetics including the reaction mechanisms of hydrogen and methane combustion. We also study how the inadequacy representation affects an unobserved quantity of interest— the flamespeed of a one-dimensional hydrogen laminar flame.