Monte Carlo Neural PDE Solver

Rui Zhang, Qi Meng, Rongchan Zhu, Yue Wang, Wenlei Shi, Shihua Zhang, Zhi-Ming Ma, Tie-Yan Liu

Keywords: Neural PDE Solver, Feynman-Kac Formula, AI for PDE

TL;DR: We propose MCNP Solver for training unsupervised neural PDE solvers via a Monte Carlo view.

Abstract: Training neural PDE solver in an unsupervised manner is essential in scenarios with limited available or high-quality data. However, the performance and efficiency of existing methods are limited by the properties of numerical algorithms integrated during the training stage (like FDM and PSM), which require careful spatiotemporal discretization to obtain reasonable accuracy, especially in cases with high-frequency components and long periods. To overcome these limitations, we propose Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via a Monte Carlo view, which regards macroscopic phenomena as ensembles of random particles. MCNP Solver naturally inherits the advantages of the Monte Carlo method (MCM), which is robust against spatial-temporal variations and can tolerate coarse time steps compared to other unsupervised methods. In practice, we develop one-step rollout and Fourier Interpolation techniques that help reduce computational costs or errors arising from time and space, respectively. Furthermore, we design a multi-scale framework to improve performance in long-time simulation tasks. In theory, we characterize the approximation error and robustness of the MCNP Solver on convection-diffusion equations. Numerical experiments on diffusion and Navier-Stokes equations demonstrate significant accuracy improvements compared to other unsupervised baselines in cases with highly variable fields and long-time simulation settings.

Submission Number: 2692