Operator learning is a rapidly growing field that aims to approximate nonlinear operators related to partial differential equations (PDEs) using neural operators. These rely on discretization of input and output functions and are, usually, expensive to train for large-scale problems at high-resolution. Motivated by this, we present a Multi-Level Monte Carlo (MLMC) approach to train neural operators by leveraging a hierarchy of resolutions of function discretization. Our framework relies on using gradient corrections from fewer samples of fine-resolution data to decrease the computational cost of training while maintaining a high level accuracy. The proposed MLMC training procedure can be applied to any architecture accepting multi-resolution data. Our numerical experiments on a range of state-of-the-art models and test-cases demonstrate improved computational efficiency compared to traditional single-resolution training approaches, and highlight the existence of a Pareto curve between accuracy and computational time, related to the number of samples per resolution.

翻译：算子学习是一个快速发展的领域，旨在利用神经算子逼近与偏微分方程相关的非线性算子。这些方法依赖于输入和输出函数的离散化，且通常在高分辨率下针对大规模问题进行训练时计算成本高昂。受此启发，我们提出了一种多级蒙特卡洛方法，通过利用函数离散化的多级分辨率层次来训练神经算子。该框架利用来自少量高分辨率数据样本的梯度修正，在保持高精度水平的同时降低训练的计算成本。所提出的MLMC训练程序可应用于任何接受多分辨率数据的架构。我们在多种先进模型和测试案例上的数值实验表明，与传统单分辨率训练方法相比，该方法显著提升了计算效率，并揭示了精度与计算时间之间存在帕累托曲线关系，该关系与每级分辨率的样本数量相关。