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 based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction-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 correction scheme. Further, topology optimization involving a nonlinear reaction-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.

翻译：本文聚焦于通过神经算子逼近一类参数化偏微分方程解算子方法的发展。神经算子面临多项挑战，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、优化与控制等下游问题中的应用。基于先前在JCP 486 (2023) 112104中的工作，本文考虑了一个基于线性变分问题框架，该框架可对神经算子的预测结果进行修正。我们进一步分析了与该修正问题相关的算子，称为基于残差的误差修正算子（简称修正算子）。基于PCANet型神经算子的二维非线性反应扩散模型数值结果表明，采用修正方案后，近似解的精度提升了近两个数量级。此外，我们针对含非线性反应扩散模型的拓扑优化问题展开研究，凸显了神经算子的局限性及修正方案的有效性。使用神经算子代理的优化器会出现显著误差（高达80%），而经修正的神经算子其误差则大幅降低（低于7%）。