An Uncertainty-Aware, Mesh-Free Numerical Method for Kolmogorov PDEs

Abstract This study introduces an uncertainty-aware, mesh-free numerical method for solving Kolmogorov PDEs. In the proposed method, we use Gaussian process regression (GPR) to smoothly interpolate pointwise solutions that are obtained by Monte Carlo methods based on the Feynman–Kac formula. The proposed method has two main advantages: 1. uncertainty assessment, which provides numerical information about the validity of the solution, and 2. mesh-free computation, which allows one to choose any low-dimensional bounded domain for reducing the computational cost. The quality of the solution is improved by adjusting the kernel function and incorporating noise information from the Monte Carlo samples into the GPR noise model. The performance of the method is rigorously analyzed based on a theoretical lower bound on the posterior variance, which serves as a measure of the error between the numerical and true solutions. Extensive tests on three representative PDEs demonstrate the high accuracy and robustness of the method compared to existing methods.