# A Gaussian Process Modeling Approach for Predicting the Solution of a Partial Differential Equation

Project: Research › GRF

## Researcher(s)

## Description

This project will develop a probabilistic method based on Gaussian process (GP) models to predict the solution of a partial differential equation (PDE) as a function of its inputs. A PDE, together with its inputs (spatial domain, equation parameters, and boundary and initial conditions), gives a prediction of one or more physical quantities over a space-time domain as its solution. For many practical problems, there is a need to quantify changes in the solution due to changes in the inputs. For instance, there is growing research in the subject of uncertainty quantification (UQ), which includes the forward problem of quantifying uncertainty in functionals of the PDE solution due to uncertainty in the inputs, and also the inverse problem of estimating unknown PDE inputs. Solving PDEs to high accuracy at many input settings is time-consuming. Thus, a statistical model that allows less accurate prediction of the solution with less computational effort but rigorous quantification of the prediction error is a useful alternative. This project will develop a GP model with these properties. It gives a probabilistic prediction of the true PDE solution as a function of the inputs using information from the PDE and its boundary conditions at a finite set of points. The proposed GP model is a nonparametric Bayesian approach to estimate the true PDE solution over an input domain. The effect of numerical errors in solving PDEs with meshless collocation methods is often ignored. Our model, which is a probabilistic meshless collocation method, will be used to quantify these errors and to develop sequential design approaches to solve the unsolved problem of balancing the quality of the discretization in the space-time and input domains. The proposed model has potentially pioneering applications in metamodel construction, parameter calibration, and product optimization. At present, discretization error in the space-time domain is ignored in metamodel building via computer experiments, i.e., the numerical solution is assumed to be the true solution, and effort is focused on quantifying the discretization error in the input domain. In parameter calibration, attempts to account for model bias lead to non-identifiable statistical models. Our model eliminates bias due to discretization error in the space-time domain. In product optimization, time-consuming high accuracy finite element simulations are performed to eliminate numerical noise that makes optimization of some functionals of the PDE solution difficult. Our model produces smooth estimates of these functionals without utilizing a fine discretization in the space-time domain.## Detail(s)

Status | Active |
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Effective start/end date | 1/01/18 → … |