Gaussian process for estimating parameters of partial differential equations and its application to the Richards equation

This paper proposes a new collocation method for estimating parameters of a partial differential equation (PDE), which uses Gaussian process (GP) as a basis function and is termed as Gaussian process for partial differential equation (GPPDE). The conventional method of estimating parameters of a differential equation is to minimize the error between observations and their estimates. The estimates are produced from the forward solution (numerical or analytical) of the differential equation. The conventional approach requires initial and boundary conditions, and discretization of differential equations if the forward solution is obtained numerically. The proposed method requires fitting a GP regression model to the observations of the state variable, then obtaining derivatives of the state variable using the property that derivative of a GP is also a GP, and finally adjusting the PDE parameters so that the GP derived partial derivatives satisfy the PDE. The method does not require initial and boundary conditions, however if these conditions are available (exactly or with measurement errors), they can be easily incorporated. The GPPDE method is evaluated by applying it on the diffusion and the Richards equations. The results suggest that GPPDE can correctly estimate parameters of the two equations. For the Richards equation, GPPDE performs well in the presence of noise. A comparison of GPPDE with HYDRUS-1D software showed that their performances are comparable, though GPPDE has significant advantages in terms of computational time. GPPDE could be an effective alternative to conventional approaches for finding parameters of high-dimensional PDEs where large datasets are available.