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Neural operators as novel neural architectures for fast approximatingsolution operators of partial differential equations (PDEs), have shownconsiderable promise for future scientific computing. However, the mainstreamof training neural operators is still data-driven, which needs an expensiveground-truth dataset from various sources (e.g., solving PDEs' samples with theconventional solvers, real-world experiments) in addition to training stagecosts. From a computational perspective, marrying operator learning andspecific domain knowledge to solve PDEs is an essential step in reducingdataset costs and label-free learning. We propose a novel paradigm thatprovides a unified framework of training neural operators and solving PDEs withthe variational form, which we refer to as the variational operator learning(VOL). Ritz and Galerkin approach with finite element discretization aredeveloped for VOL to achieve matrix-free approximation of system functional andresidual, then direct minimization and iterative update are proposed as twooptimization strategies for VOL. Various types of experiments based onreasonable benchmarks about variable heat source, Darcy flow, and variablestiffness elasticity are conducted to demonstrate the effectiveness of VOL.With a label-free training set and a 5-label-only shift set, VOL learnssolution operators with its test errors decreasing in a power law with respectto the amount of unlabeled data. To the best of the authors' knowledge, this isthe first study that integrates the perspectives of the weak form and efficientiterative methods for solving sparse linear systems into the end-to-endoperator learning task.

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