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

Abstract

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.

arXiv comment36 pages, 14 figures, 3 tables

References in corpus (19)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems
• Simulation-based optimal Bayesian experimental design for nonlinear systems
• Physics-Informed Neural Operator for Learning Partial Differential Equations
• Operator inference for non-intrusive model reduction with quadratic manifolds
• Optimal design of acoustic metamaterial cloaks under uncertainty
• Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty
• Modeling and simulation of vascular tumors embedded in evolving capillary networks
• Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
• Wavelet neural operator: a neural operator for parametric partial differential equations
• Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network
• Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks
• Numerical analysis of nonlocal fracture models in Hölder space
• A globally convergent modified Newton method for the direct minimization of the Ohta-Kawasaki energy with application to the directed self-assembly of diblock copolymers
• Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators
• Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithm
• High-dimensional Nonlinear Bayesian Inference of Poroelastic Fields from Pressure Data
• Goal-Oriented A-Posteriori Estimation of Model Error as an Aid to Parameter Estimation
• Hessian-informed Hamiltonian Monte Carlo for high-dimensional problems