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.

基于变分方法，提出一种新的培训神经网络算子和解决偏微分方程的统一框架，称为变分算子学习（VOL），VOL 可以以近乎无标签的方式有效地学习 PDE 的解算子，并利用最陡下降法和共轭梯度法进行更新。