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

Project: Research

## Researcher(s)

## Description

This project will develop a probabilistic method based on Gaussian process (GP) modelsto 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 andinitial conditions), gives a prediction of one or more physical quantities over a space-timedomain as its solution. For many practical problems, there is a need to quantifychanges in the solution due to changes in the inputs. For instance, there is growingresearch in the subject of uncertainty quantification (UQ), which includes the forwardproblem of quantifying uncertainty in functionals of the PDE solution due to uncertaintyin the inputs, and also the inverse problem of estimating unknown PDE inputs. SolvingPDEs to high accuracy at many input settings is time-consuming. Thus, a statisticalmodel that allows less accurate prediction of the solution with less computational effortbut rigorous quantification of the prediction error is a useful alternative. This projectwill develop a GP model with these properties. It gives a probabilistic prediction of thetrue PDE solution as a function of the inputs using information from the PDE and itsboundary conditions at a finite set of points.The proposed GP model is a nonparametric Bayesian approach to estimate the true PDEsolution over an input domain. The effect of numerical errors in solving PDEs withmeshless collocation methods is often ignored. Our model, which is a probabilisticmeshless collocation method, will be used to quantify these errors and to developsequential design approaches to solve the unsolved problem of balancing the quality ofthe discretization in the space-time and input domains. The proposed model haspotentially pioneering applications in metamodel construction, parameter calibration, andproduct optimization. At present, discretization error in the space-time domain isignored in metamodel building via computer experiments, i.e., the numerical solution isassumed to be the true solution, and effort is focused on quantifying the discretizationerror in the input domain. In parameter calibration, attempts to account for model biaslead to non-identifiable statistical models. Our model eliminates bias due todiscretization error in the space-time domain. In product optimization, time-consuminghigh accuracy finite element simulations are performed to eliminate numerical noisethat makes optimization of some functionals of the PDE solution difficult. Our modelproduces smooth estimates of these functionals without utilizing a fine discretization inthe space-time domain.## Detail(s)

Project number | 9042475 |
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Grant type | GRF |

Status | Active |

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