  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.
