mlsanityData to 14 Oct 2026

Physics-Informed Neural Operator for Learning Partial Differential Equations

arXiv 2111.03794 ↗First posted 6 Nov 2021cs.LG cs.NA math.NAcited by 527

Not on the map. Why?

Cited by 22 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, i.e., being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization-convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, e.g., in multi-scale dynamic systems such as Kolmogorov flows.

Key facts. Cited by 527 papers in the map (data through 14 Oct 2026). Builds on 29 of its 66 references that are in the map. First posted on arXiv 6 Nov 2021.

Builds on

29 of 66 references.

References · 29

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  4. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  5. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  6. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  7. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  8. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  9. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  10. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  11. 2021Machine learning accelerated computational fluid dynamicscited by 471
  12. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
17 more
  1. 2019Machine Learning for Fluid Mechanicscited by 385
  2. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  3. 2017PDE-Net: Learning PDEs from Datacited by 311
  4. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  5. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  6. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  7. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 204
  8. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
  9. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  10. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  11. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 67
  12. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  13. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  14. 202086 PFLOPS Deep Potential Molecular Dynamics simulation of 100 million atoms with ab initio accuracycited by 15
  15. 2019Variational training of neural network approximations of solution maps for physical modelscited by 8
  16. 2021HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networkscited by 5
  17. 2021Data-informed Deep Optimizationcited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 38; 2023: 69; 2024: 115; 2025: 134; 2026: 171.

527 papers.

Sort:

By area:

  1. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  2. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  3. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  4. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 303
  5. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  6. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  7. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  8. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  9. 2022FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operatorscited by 188
  10. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  11. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 152
  12. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 151
38 more
  1. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 131
  2. 2023Scalable Transformer for PDE Surrogate Modelingcited by 128
  3. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  4. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 98
  5. 2022Clifford Neural Layers for PDE Modelingcited by 92
  6. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 91
  7. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 84
  8. 2022Tutorial on amortized optimizationcited by 83
  9. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
  10. 2023A Mathematical Guide to Operator Learningcited by 76
  11. 2023Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEscited by 68
  12. 2025InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciencescited by 68
  13. 2024Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networkscited by 67
  14. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  15. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 65
  16. 2024Neural Operators with Localized Integral and Differential Kernelscited by 62
  17. 2024On conditional diffusion models for PDE simulationscited by 60
  18. 2023Learning Physical Models that Can Respect Conservation Lawscited by 59
  19. 2025AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformerscited by 53
  20. 2022Generative Adversarial Neural Operatorscited by 52
  21. 2023CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamicscited by 46
  22. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  23. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  24. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 38
  25. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 38
  26. 2024A Library for Learning Neural Operatorscited by 38
  27. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  28. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 36
  29. 2025Walrus: A Cross-Domain Foundation Model for Continuum Dynamicscited by 36
  30. 2024Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solverscited by 35
  31. 2022Learning differentiable solvers for systems with hard constraintscited by 34
  32. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  33. 2022Applications of physics informed neural operatorscited by 33
  34. 2024Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learningcited by 32
  35. 2025PDE-Transformer: Efficient and Versatile Transformers for Physics Simulationscited by 32
  36. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 31
  37. 2024On the Benefits of Memory for Modeling Time-Dependent PDEscited by 31
  38. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 30

50 most cited of 527 All 527 (JSON)