Solving Fredholm integral equation of the first kind using Gaussian process regression

摘要

Fredholm integral equation of the first kind is a typical ill-posed problem, and it is usually difficult to obtain a stable numerical solution. In this paper, a new method is proposed to solve Fredholm integral equation using Gaussian process regression (GPR). The key to this method is that the right-hand term of the original integral equation is reconstructed by the GPR model to obtain a new integral equation in a reproducing kernel Hilbert spaces (RKHS). We present an analytical approximate solution of the new equation and prove that it converges to the exact minimal-norm solution of the original equation under the L2-norm. Especially, for the degenerate kernel equation, we obtain an explicit formula of the exact minimal-norm solution. Finally, the proposed method is verified to be very effective in solution accuracy by multiple examples.