mlsanityData to 7 Oct 2026

Physics-Informed Deep Neural Operator Networks

arXiv 2207.05748 ↗First posted 8 Jul 2022cs.LG cs.NA math.NAcited by 110

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Abstract (arXiv)

Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if the governing equations are known they can be incorporated into the loss function during training to develop physics-informed neural operators. Neural operators can be used as surrogates in design problems, uncertainty quantification, autonomous systems, and almost in any application requiring real-time inference. Moreover, independently pre-trained DeepONets can be used as components of a complex multi-physics system by coupling them together with relatively light training. Here, we present a review of DeepONet, the Fourier neural operator, and the graph neural operator, as well as appropriate extensions with feature expansions, and highlight their usefulness in diverse applications in computational mechanics, including porous media, fluid mechanics, and solid mechanics.

Key facts. Cited by 110 papers in the map (data through 7 Oct 2026). Builds on 35 of its 59 references that are in the map. First posted on arXiv 8 Jul 2022.

Builds on

35 of 59 references.

References · 35

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2017Graph Attention Networkscited by 8,854
  3. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  9. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  10. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  11. 2021GRAND: Graph Neural Diffusioncited by 258
  12. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
23 more
  1. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  2. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  3. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  4. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  5. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  6. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  7. 2022Learning Operators with Coupled Attentioncited by 88
  8. 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 77
  9. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  10. 2018Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operatorcited by 70
  11. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  12. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  13. 2022Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problemscited by 57
  14. 2022Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problemscited by 47
  15. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
  16. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  17. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  18. 2022On the influence of over-parameterization in manifold based surrogates and deep neural operatorscited by 29
  19. 2022Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONetscited by 27
  20. 2022Scalable algorithms for physics-informed neural and graph networkscited by 21
  21. 2022A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurementscited by 19
  22. 2022Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysmscited by 14
  23. 2021Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networkscited by 12

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 13; 2024: 33; 2025: 38; 2026: 24.

110 papers.

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  1. 2023A Mathematical Guide to Operator Learningcited by 74
  2. 2023Learning Physical Models that Can Respect Conservation Lawscited by 58
  3. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  4. 2024Transformers as Neural Operators for Solutions of Differential Equations with Finite Regularitycited by 33
  5. 2024State-space models are accurate and efficient neural operators for dynamical systemscited by 27
  6. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  7. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  8. 2024Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networkscited by 20
  9. 2023Improved generalization with deep neural operators for engineering systems: Path towards digital twincited by 19
  10. 2024Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solvingcited by 17
  11. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  12. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
38 more
  1. 2025Operator Learning: A Statistical Perspectivecited by 14
  2. 2023Exactly conservative physics-informed neural networks and deep operator networks for dynamical systemscited by 13
  3. 2024Higher-order-ReLU-KANs (HRKANs) for solving physics-informed neural networks (PINNs) more accurately, robustly and fastercited by 13
  4. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
  5. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 10
  6. 2025Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEscited by 9
  7. 2025Solving Sparse Finite Element Problems on Neuromorphic Hardwarecited by 8
  8. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  9. 2025Structure-Preserving Digital Twins via Conditional Neural Whitney Formscited by 8
  10. 2025HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutionscited by 8
  11. 2023Scattering with Neural Operatorscited by 7
  12. 2022Adaptive physics-informed neural operator for coarse-grained non-equilibrium flowscited by 6
  13. 2024Physics-informed Discretization-independent Deep Compositional Operator Networkcited by 6
  14. 2024Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured gridcited by 6
  15. 2024Efficient Training of Deep Neural Operator Networks via Randomized Samplingcited by 6
  16. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  17. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  18. 2025Operator Learning at Machine Precisioncited by 6
  19. 2025Towards a Foundation Model for Partial Differential Equations Across Physics Domainscited by 6
  20. 2023Importance of equivariant and invariant symmetries for fluid flow modelingcited by 5
  21. 2023PINN surrogate of Li-ion battery models for parameter inference. Part II: Regularization and application of the pseudo-2D modelcited by 5
  22. 2024A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertaintycited by 5
  23. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  24. 2025Towards a Physics Foundation Modelcited by 5
  25. 2024Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equationscited by 4
  26. 2024Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantificationcited by 4
  27. 2024A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solidscited by 4
  28. 2025Verification and Validation for Trustworthy Scientific Machine Learningcited by 4
  29. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  30. 2026OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inferencecited by 4
  31. 2024Functional SDE approximation inspired by a deep operator network architecturecited by 3
  32. 2024Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularizationcited by 3
  33. 2024Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiologycited by 3
  34. 2025Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networkscited by 3
  35. 2025EquiNO: A Physics-Informed Neural Operator for Multiscale Simulationscited by 3
  36. 2025Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operatorscited by 3
  37. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  38. 2023Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operatorcited by 2

50 most cited of 110 All 110 (JSON)