Based on the variational method, we propose a novel paradigm that provides a unified framework of training neural operators and solving partial differential equations (PDEs) with the variational form, which we refer to as the variational operator learning (VOL). We first derive the functional approximation of the system from the node solution prediction given by neural operators, and then conduct the variational operation by automatic differentiation, constructing a forward-backward propagation loop to derive the residual of the linear system. One or several update steps of the steepest decent method (SD) and the conjugate gradient method (CG) are provided in every iteration as a cheap yet effective update for training the neural operators. Experimental results show the proposed VOL can learn a variety of solution operators in PDEs of the steady heat transfer and the variable stiffness elasticity with satisfactory results and small error. The proposed VOL achieves nearly label-free training. Only five to ten labels are used for the output distribution-shift session in all experiments. Generalization benefits of the VOL are investigated and discussed.

翻译：基于变分方法，我们提出了一种新颖范式，为训练神经算子及求解具有变分形式的偏微分方程（PDEs）提供了统一框架，称之为变分算子学习（VOL）。首先从神经算子给出的节点解预测推导系统的函数逼近，进而通过自动微分执行变分运算，构建前向-反向传播循环以导出线性系统的残差。每次迭代中采用最速下降法（SD）和共轭梯度法（CG）的一步或多步更新，作为训练神经算子的低成本高效更新策略。实验结果表明，所提出的VOL能够学习稳态热传导及变刚度弹性力学中多种PDE解算子，并获得满意结果与较小误差。该方法实现了近乎无标签训练——在所有实验的输出分布偏移环节中仅使用5至10个标签。本文还对VOL的泛化优势进行了研究与讨论。