mlsanityData to 7 Oct 2026

Guiding continuous operator learning through Physics-based boundary constraints

arXiv 2212.07477 ↗First posted 14 Dec 2022cs.LG math.AP math.OAcited by 21

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Abstract (arXiv)

Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical meaning, and guarantee the existence and the uniqueness of the PDE solution. Current neural-network based approaches that aim to solve PDEs rely only on training data to help the model learn BCs implicitly. There is no guarantee of BC satisfaction by these models during evaluation. In this work, we propose Boundary enforcing Operator Network (BOON) that enables the BC satisfaction of neural operators by making structural changes to the operator kernel. We provide our refinement procedure, and demonstrate the satisfaction of physics-based BCs, e.g. Dirichlet, Neumann, and periodic by the solutions obtained by BOON. Numerical experiments based on multiple PDEs with a wide variety of applications indicate that the proposed approach ensures satisfaction of BCs, and leads to more accurate solutions over the entire domain. The proposed correction method exhibits a (2X-20X) improvement over a given operator model in relative L2 error (0.000084 relative L2 error for Burgers' equation).

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 28 of its 52 references that are in the map. First posted on arXiv 14 Dec 2022.

Builds on

28 of 52 references.

References · 28

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  5. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  8. 2021Machine learning accelerated computational fluid dynamicscited by 461
  9. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  11. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  12. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
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  1. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 237
  2. 2021Factorized Fourier Neural Operatorscited by 237
  3. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  4. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  5. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  6. 2021DC3: A learning method for optimization with hard constraintscited by 181
  7. 2017Solving ill-posed inverse problems using iterative deep neural networkscited by 145
  8. 2019Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networkscited by 144
  9. 2022Monarch: Expressive Structured Matrices for Efficient and Accurate Trainingcited by 100
  10. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  11. 2020MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Frameworkcited by 78
  12. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  13. 2020EikoNet: Solving the Eikonal equation with Deep Neural Networkscited by 50
  14. 2019Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problemscited by 31
  15. 2018BCR-Net: a neural network based on the nonstandard wavelet formcited by 18
  16. 2019Meta-learning Pseudo-differential Operators with Deep Neural Networkscited by 17

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Citing papers per year: 2023: 4; 2024: 8; 2025: 5; 2026: 4.

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  1. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 134
  2. 2023PreDiff: Precipitation Nowcasting with Latent Diffusion Modelscited by 96
  3. 2023Learning Physical Models that Can Respect Conservation Lawscited by 58
  4. 2025CodePDE: An Inference Framework for LLM-driven PDE Solver Generationcited by 37
  5. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 35
  6. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  7. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
  8. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  9. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 22
  10. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 14
  11. 2024AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamicscited by 13
  12. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
9 more
  1. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generationcited by 11
  2. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 8
  3. 2026CompNO: A Novel Foundation Model approach for solving Partial Differential Equationscited by 4
  4. 2024Structure-Preserving Operator Learningcited by 3
  5. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  6. 2026Imposing Boundary Conditions on Neural Operators via Learned Function Extensionscited by 3
  7. 2025Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operatorscited by 1
  8. 2026Enforcing Dirichlet Boundary Conditions in Operator Learningcited by 1
  9. 2026Generalized Neural Operator for Parametric and Boundary-Value Problemscited by 0