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.

翻译：神经算子作为一种新型神经架构，能够快速近似偏微分方程的解算子，在未来的科学计算领域展现出巨大潜力。然而，目前主流的神经算子训练仍依赖数据驱动方式，除了训练阶段的成本外，还需要昂贵的真实标注数据集（例如通过传统求解器采集偏微分方程样本、进行真实实验等）。从计算角度而言，将算子学习与特定领域知识相结合来求解偏微分方程，是降低数据集成本并实现无标签学习的关键步骤。我们提出了一种新范式——变分算子学习（VOL），该范式将训练神经算子与求解变分形式的偏微分方程统一在同一个框架中。基于有限元离散的Ritz和Galerkin方法被用于实现VOL系统泛函和残差的无矩阵近似，随后提出直接最小化和迭代更新两种优化策略。通过基于变热源、达西流和变刚度弹性力学等合理基准的多类型实验，验证了VOL的有效性。在无标签训练集和仅含5个标签的偏移集条件下，VOL学习解算子的测试误差随无标签数据量呈幂律递减。据作者所知，这是首次将弱形式理论与稀疏线性系统高效迭代方法的视角整合至端到端算子学习任务中的研究。