This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework is proposed based on the linear variational problem that gives the correction to the prediction furnished by neural operators. The operator associated with the corrector problem is referred to as the corrector operator. Numerical results involving a nonlinear diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the proposed scheme. Further, topology optimization involving a nonlinear diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected following the proposed method.

翻译：本文聚焦于通过神经算子近似一类参数化偏微分方程解算子的方法开发。神经算子面临若干挑战，包括生成合适训练数据的问题、成本-精度权衡以及非平凡的超参数调优。神经算子精度的不可预测性影响了其在推理、优化和控制等下游问题中的应用。本文提出一个基于线性变分问题的框架，用于修正神经算子提供的预测。与该修正问题相关的算子被称为校正算子。采用PCANet型神经算子的二维非线性扩散模型的数值结果表明，使用所提方案对神经算子进行校正后，近似精度提升近两个数量级。此外，本文还考虑涉及非线性扩散模型的拓扑优化，以突出神经算子的局限性及校正方案的有效性。采用神经算子代理的优化器会出现显著误差（高达80%），而使用所提方法校正神经算子后，误差大幅降低（低于7%）。