mlsanityData to 14 Oct 2026

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

arXiv 2306.12047 ↗First posted 21 Jun 2023math.NA cs.LG cs.NAcited by 9

Not on the map. Why?

Abstract (arXiv)

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.

Key facts. Cited by 9 papers in the map (data through 14 Oct 2026). Builds on 27 of its 102 references that are in the map. First posted on arXiv 21 Jun 2023.

Builds on

27 of 102 references.

References · 27

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  5. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  6. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  7. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  8. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 297
  9. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  10. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  11. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  12. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
15 more
  1. 2020Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEscited by 40
  2. 2021Reduced operator inference for nonlinear partial differential equationscited by 21
  3. 2022Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problemscited by 21
  4. 2013A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problemscited by 19
  5. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 19
  6. 2022Large-scale Bayesian optimal experimental design with derivative-informed projected neural networkcited by 17
  7. 2021Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networkscited by 16
  8. 2020Optimal design of acoustic metamaterial cloaks under uncertaintycited by 7
  9. 2022Bayesian Inference of Tissue Heterogeneity for Individualized Prediction of Glioma Growthcited by 6
  10. 2018Taylor approximation and variance reduction for PDE-constrained optimal control under uncertaintycited by 5
  11. 2016Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximationscited by 4
  12. 2022Goal-Oriented A-Posteriori Estimation of Model Error as an Aid to Parameter Estimationcited by 4
  13. 2020A globally convergent modified Newton method for the direct minimization of the Ohta-Kawasaki energy with application to the directed self-assembly of diblock copolymerscited by 2
  14. 2020Peridynamics-based discrete element method (PeriDEM) model of granular systems involving breakage of arbitrarily shaped particlescited by 1
  15. 2022Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithmcited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 4; 2025: 2; 2026: 3.

9 papers.

  1. 2025PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulationscited by 8
  2. 2025Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learningcited by 7
  3. 2024A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertaintycited by 5
  4. 2024Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equationscited by 4
  5. 2024A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domainscited by 3
  6. 2024Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approachcited by 2
  7. 2026Error-Conditioned Neural Solverscited by 1
  8. 2026Structured Extrema Errors in Classical Surrogates for Viscous Burgers: A Physics-Consistent Interpretationcited by 0
  9. 2026Adapting neural operators for mechanics decisions under changing operating conditionscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Stability and Discretization Error of State Space Model Neural Operators
  2. 2025Mixture of neural operator experts for learning boundary conditions and model selection
  3. 2025Residual-augmented flow matching operators for probabilistic partial differential equations
  4. 2025Latent Mamba Operator for Partial Differential Equations
  5. 2026Learning Physical Operators using Neural Operators
  6. 2026Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations
  7. 2024Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs
  8. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems