In scenarios with limited available or high-quality 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 random walk 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 memory and time issues associated with Monte Carlo sampling, offering an improvement over traditional Monte Carlo methods. 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.

翻译：在有限或低质量数据场景下，以无监督方式训练函数到函数的神经偏微分方程（PDE）求解器至关重要。然而，现有方法的效率和精度受限于训练阶段集成的数值算法特性（如有限差分法和伪谱法）。这些方法需精细的时空离散化才能达到合理精度，导致显著的计算挑战和不准确的模拟，尤其在大时空变化场景中。为解决这些限制，我们提出蒙特卡洛神经PDE求解器（MCNP Solver），通过PDE的概率表示训练无监督神经求解器，将宏观现象视为随机粒子的集合。与其他无监督方法相比，MCNP Solver天然继承蒙特卡洛方法的优势：对时空变化具有鲁棒性且能容忍粗大步长。在模拟粒子随机游走过程中，我们对平流过程采用Heun方法，并通过扩散过程中相邻网格点的概率密度函数计算期望。这些技术提升了精度，规避了蒙特卡洛采样相关的计算内存与时间问题，是对传统蒙特卡洛方法的改进。在对流-扩散方程、Allen-Cahn方程和Navier-Stokes方程的数值实验中，相比其他无监督基线方法，本方法在精度和效率上均有显著提升。源代码将公开发布于：https://github.com/optray/MCNP。