Multi-Level Monte Carlo Training of Neural Operators
This addresses a computational bottleneck in operator learning for PDEs, offering an incremental improvement in training efficiency.
The paper tackles the high computational cost of training neural operators for PDEs by introducing a Multi-Level Monte Carlo (MLMC) training approach that uses a hierarchy of resolutions, achieving improved computational efficiency while maintaining accuracy.
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 dicretization. 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.