Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously.

Automated summary

This paper investigates the parameter estimation problem for various types of differential equations controlled by linear operators. Data-driven algorithms based on Gaussian processes are employed to solve the inverse problem and estimate the unknown parameters of the partial differential equations. Numerical tests demonstrate that the data-driven methods based on Gaussian processes can accurately estimate the parameters and approximate the solutions and inhomogeneous terms of the considered partial differential equations simultaneously.