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In scenarios with limited available or high-quality data, training thefunction-to-function neural PDE solver in an unsupervised manner is essential.However, the efficiency and accuracy of existing methods are constrained by theproperties of numerical algorithms, such as finite difference andpseudo-spectral methods, integrated during the training stage. These methodsnecessitate careful spatiotemporal discretization to achieve reasonableaccuracy, leading to significant computational challenges and inaccuratesimulations, 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 ensemblesof random particles. Compared to other unsupervised methods, MCNP Solvernaturally inherits the advantages of the Monte Carlo method, which is robustagainst spatiotemporal variations and can tolerate coarse step size. Insimulating the random walk of particles, we employ Heun's method for theconvection process and calculate the expectation via the probability densityfunction of neighbouring grid points during the diffusion process. Thesetechniques enhance accuracy and circumvent the computational memory and timeissues associated with Monte Carlo sampling, offering an improvement overtraditional Monte Carlo methods. Our numerical experiments onconvection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstratesignificant improvements in accuracy and efficiency compared to otherunsupervised baselines. The source code will be publicly available at:https://github.com/optray/MCNP.

Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation