An Uncertainty-aware, Mesh-free Computational Method for Partial Differential Equations

A study conducted by Daisuke Inoue et al., in collaboration with the University of Tokyo, was published in the Journal of Scientific Computing.
This study proposes a new mesh-free numerical method for solving Kolmogorov partial differential equations (PDEs). The method uses the Feynman-Kac formula-based Monte Carlo approach to obtain pointwise solutions, which are then smoothly interpolated using Gaussian Process Regression (GPR). This approach offers the advantages of uncertainty quantification for solution validity and reduced computational cost through mesh-free computation. The method's accuracy is evaluated based on the theoretical lower bound of posterior variance, and extensive tests demonstrate its high precision and robustness.

Title: An Uncertainty-aware, Mesh-free Numerical Method for Kolmogorov PDEs
Authors: Inoue, D., Ito, Y., Kashiwabara, T., Saito, N., Yoshida, H.
Journal Name: Journal of Scientific Computing
Published: March 24, 2025
https://doi.org/10.1007/s10915-025-02846-9