Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation

TL;DR

This paper introduces the Monte Carlo Neural PDE Solver (MCNP Solver), a novel unsupervised neural network approach that leverages probabilistic particle representations to improve the accuracy and efficiency of solving PDEs with limited data.

Contribution

The paper proposes a Monte Carlo-based neural PDE solver that is robust to spatiotemporal variations and improves upon existing unsupervised methods in accuracy and computational efficiency.

Findings

Demonstrates superior accuracy on convection-diffusion, Allen-Cahn, and Navier-Stokes equations.

Shows robustness to coarse discretizations and large spatiotemporal variations.

Achieves significant efficiency gains over traditional numerical methods.

Abstract

In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the trajectories of particles, we employ Heun's method for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines. The source code will be publicly available at: https://github.com/optray/MCNP.