Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations

Abstract

Neural operators as novel neural architectures for fast approximating solution operators of partial differential equations (PDEs), have shown considerable promise for future scientific computing. However, the mainstream of training neural operators is still data-driven, which needs an expensive ground-truth dataset from various sources (e.g., solving PDEs’ samples with the conventional solvers, real-world experiments) in addition to training stage costs. From a computational perspective, marrying operator learning and specific domain knowledge to solve PDEs is an essential step for data-efficient and low-carbon learning. We propose a novel data-efficient paradigm that provides a unified framework of training neural operators and solving PDEs with the domain knowledge related to the variational form, which we refer to as the variational operator learning (VOL). We develop Ritz and Galerkin approach respectively with finite element discretization for VOL to achieve matrix-free approximation of the energy functional of physical systems and calculation of residual tensors derived from associated linear systems with linear time complexity and O(1) space complexity. We then propose direct minimization and iterative update as two possible optimization strategies. Various types of experiments based on reasonable benchmarks about variable heat source, Darcy flow, and variable stiffness elasticity are conducted to demonstrate the effectiveness of VOL. With a label-free training set, VOL learns solution operators with its test errors decreasing in a power law with respect to the amount of unlabeled data. To the best of the authors’ knowledge, this is the first study that integrates the perspectives of the weak form and efficient iterative methods for solving sparse linear systems into the end-to-end operator learning task. Codes and Datasets are publicly available at https://doi.org/10.5281/zenodo.11097870.