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This work focuses on developing methods for approximating the solutionoperators of a class of parametric partial differential equations via neuraloperators. Neural operators have several challenges, including the issue ofgenerating appropriate training data, cost-accuracy trade-offs, and nontrivialhyperparameter tuning. The unpredictability of the accuracy of neural operatorsimpacts their applications in downstream problems of inference, optimization,and control. A framework based on the linear variational problem that gives thecorrection to the prediction furnished by neural operators is considered basedon earlier work in JCP 486 (2023) 112104. The operator, called Residual-basedError Corrector Operator or simply Corrector Operator, associated with thecorrector problem is analyzed further. Numerical results involving a nonlinearreaction-diffusion model in two dimensions with PCANet-type neural operatorsshow almost two orders of increase in the accuracy of approximations whenneural operators are corrected using the correction scheme. Further, topologyoptimization involving a nonlinear reaction-diffusion model is considered tohighlight the limitations of neural operators and the efficacy of thecorrection scheme. Optimizers with neural operator surrogates are seen to makesignificant errors (as high as 80 percent). However, the errors are much lower(below 7 percent) when neural operators are corrected.

computer methods in applied mechanics and engineering

Residual-Based Error Corrector Operator to Enhance Accuracy and  Reliability of Neural Operator Surrogates of Nonlinear Variational  Boundary-Value Problems