Neural operators, which use deep neural networks to approximate the solution mappings of partial differential equation (PDE) systems, are emerging as a new paradigm for PDE simulation. The neural operators could be trained in supervised or unsupervised ways, i.e., by using the generated data or the PDE information. The unsupervised training approach is essential when data generation is costly or the data is less qualified (e.g., insufficient and noisy). However, its performance and efficiency have plenty of room for improvement. To this end, we design a new loss function based on the Feynman-Kac formula and call the developed neural operator Monte-Carlo Neural Operator (MCNO), which can allow larger temporal steps and efficiently handle fractional diffusion operators. Our analyses show that MCNO has advantages in handling complex spatial conditions and larger temporal steps compared with other unsupervised methods. Furthermore, MCNO is more robust with the perturbation raised by the numerical scheme and operator approximation. Numerical experiments on the diffusion equation and Navier-Stokes equation show significant accuracy improvement compared with other unsupervised baselines, especially for the vibrated initial condition and long-time simulation settings.

翻译：神经算子利用深度神经网络逼近偏微分方程系统的解映射，正成为偏微分方程仿真的新范式。此类神经算子可通过监督或无监督方式进行训练，即利用生成数据或偏微分方程信息进行学习。当数据生成成本高昂或数据质量不足（如数据匮乏或含噪声）时，无监督训练方法尤为关键。然而其性能与效率仍有较大提升空间。为此，我们基于Feynman-Kac公式设计了一种新损失函数，并将所发展的神经算子命名为蒙特卡洛神经算子（MCNO），该算子可兼容更大的时间步长并高效处理分数阶扩散算子。分析表明，与其他无监督方法相比，MCNO在处理复杂空间条件及更大时间步长方面具有优势。此外，MCNO对数值格式和算子近似引起的扰动具有更强的鲁棒性。在扩散方程和Navier-Stokes方程上的数值实验表明，相较于其他无监督基线方法，MCNO在精度上取得显著提升，尤其适用于振动初始条件和长时间仿真场景。