Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems

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 f urnished 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, topolog y 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.

Automated summary

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations using neural operators. It addresses challenges such as generating appropriate training data, cost-accuracy trade-offs, and hyperparameter tuning. The study introduces a framework based on a linear variational problem to correct the predictions generated by neural operators, and analyzes the associated correction operator. Numerical results demonstrate a significant increase in approximation accuracy when neural operators are corrected using the proposed scheme. Additionally, the limitations of neural operators and the efficacy of the correction scheme are highlighted in a topology optimization problem.