  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 is facilitated by the
probabilistic nature of GPR, and 2. mesh-free computation, which allows
efficient handling of high-dimensional PDEs. 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.
