An Uncertainty-aware, Mesh-free Numerical Method for Kolmogorov PDEs
Daisuke Inoue, Yuji Ito, Takahito Kashiwabara, Norikazu Saito, Hiroaki Yoshida
TL;DR
The paper tackles solving high-dimensional Kolmogorov PDEs while quantifying numerical uncertainty. It presents an uncertainty-aware, mesh-free approach that couples Feynman–Kac Monte Carlo sampling with heteroscedastic Gaussian Process Regression, enabling both a posterior mean solution and a posterior variance-based uncertainty measure. A theoretical contribution provides an a priori lower bound on the posterior variance (IMSE) and a probabilistic bound on the noise estimate, guiding performance expectations. Empirical results on heat, advection–diffusion, and HJB equations in 10 dimensions show improved accuracy and reduced uncertainty, validating the method against standard GPR and linear interpolation. The work advances uncertainty-aware numerical PDEs and suggests adaptive, information-driven sampling to further enhance efficiency.
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 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.