mlsanityData to 7 Oct 2026

Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML

arXiv 2301.13331 ↗First posted 30 Jan 2023cs.AI cs.LG physics.comp-phcited by 8

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Abstract (arXiv)

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Difference Methods (FDMs) require considerable time and are computationally expensive. In contrast, data-driven machine learning-based methods, such as neural networks, provide a faster, fairly accurate alternative, and, in particular, focus on neural operators, which have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the open problems of machine learning-based approaches. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

Key facts. Cited by 8 papers in the map (data through 7 Oct 2026). Builds on 45 of its 128 references that are in the map. First posted on arXiv 30 Jan 2023.

Builds on

45 of 128 references.

References · 45

  1. 2017Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networkscited by 4,861
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  5. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  6. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
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33 more
  1. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  2. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  4. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  6. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  7. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 237
  8. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  9. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  10. 2022Fast Sampling of Diffusion Models via Operator Learningcited by 173
  11. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  12. 2022AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutionscited by 77
  13. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  14. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  15. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 56
  16. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
  17. 2022Generative Adversarial Neural Operatorscited by 51
  18. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  19. 2022Applications of physics informed neural operatorscited by 33
  20. 2022On the influence of over-parameterization in manifold based surrogates and deep neural operatorscited by 29
  21. 2021Differentiable sampling of molecular geometries with uncertainty-based adversarial attackscited by 25
  22. 2022Variable-Input Deep Operator Networkscited by 24
  23. 2022NTFields: Neural Time Fields for Physics-Informed Robot Motion Planningcited by 22
  24. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  25. 2020A Sample Complexity Separation between Non-Convex and Convex Meta-Learningcited by 17
  26. 2022Graph Neural Networks in Particle Physics: Implementations, Innovations, and Challengescited by 12
  27. 2022Use of Multifidelity Training Data and Transfer Learning for Efficient Construction of Subsurface Flow Surrogate Modelscited by 11
  28. 2023Forecasting subcritical cylinder wakes with Fourier Neural Operatorscited by 11
  29. 2021Fourier Neural Operator Networks: A Fast and General Solver for the Photoacoustic Wave Equationcited by 10
  30. 2019On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Mapscited by 8
  31. 2022PaCMO: Partner Dependent Human Motion Generation in Dyadic Human Activity using Neural Operatorscited by 8
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  33. 2021Residual fourier neural operator for thermochemical curing of compositescited by 4

Built on by

Cited by year (2026 to 7 Oct)

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

8 papers.

  1. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  2. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  3. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  4. 2026Topological Neural Operatorscited by 2
  5. 2024Fourier neural operators for spatiotemporal dynamics in two-dimensional turbulencecited by 1
  6. 2026Operator Learning Using Weak Supervision from Walk-on-Spherescited by 1
  7. 2025HessianForge: Scalable LiDAR reconstruction with Physics-Informed Neural Representation and Smoothness Energy Constraintscited by 0
  8. 2025Neuro-Symbolic Operator for Interpretable and Generalizable Characterization of Complex Piezoelectric Systemscited by 0