Machine Learning for Quantifying and Reducing Model Uncertainty in Non-Linear Partial Differential Equation Models

We propose a new approach for uncertainty quantification and reduction in partial differential equation (PDE) models using data and machine learning (ML) techniques. Physical models are typically based on conservation laws. Uncertainty in such models arises from constitutive relationships (e.g., Newtonian stress model in the Navier-Stokes equation and Fick?s Law in the diffusion equation) required to close the system of PDEs. Usually, constitutive relationships are obtained by fitting an assumed function or differential operator to data. Such approach does not directly take into account uncertainty in data and uncertainty due to misfit between the constitutive model and data. Instead, we use a Gaussian Process (GP) model, a statistical machine learning method, to build a probabilistic constitutive model from data. A combination of a stochastic constitutive model with the corresponding conservation law equations results in a system of stochastic PDEs. Standard UQ techniques, including Monte Carlo, Polynomial Chaos, and Moment Equations, can be used to solve these PDEs. We use the Moment Equations Method to compute mean (prediction) and covariance (measure of model uncertainty) of state variables. By construction, the covariance function satisfies the conservation laws and initial and boundary conditions. We demonstrate that, when the measurements of state variables are available, this covariance function yields a more accurate Gaussian Process model of state variables than a purely data-driven GP model.