  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 in reducing
dataset costs and label-free learning. We propose a novel paradigm that
provides a unified framework of training neural operators and solving PDEs with
the variational form, which we refer to as the variational operator learning
(VOL). Ritz and Galerkin approach with finite element discretization are
developed for VOL to achieve matrix-free approximation of system functional and
residual, then direct minimization and iterative update are proposed as two
optimization strategies for VOL. 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 and a 5-label-only shift 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.
