mlsanityData to 14 Oct 2026

Operator Learning with Neural Fields: Tackling PDEs on General Geometries

arXiv 2306.07266 ↗First posted 12 Jun 2023cs.LG cs.AIcited by 84

Not on the map. Why?

Abstract (arXiv)

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

Key facts. Cited by 84 papers in the map (data through 14 Oct 2026). Builds on 39 of its 52 references that are in the map. First posted on arXiv 12 Jun 2023.

Builds on

39 of 52 references.

References · 39

  1. 2017Attention Is All You Needcited by 60,361
  2. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  3. 2020An Image is Worth 16x16 Words: Transformers for Image Recognition at Scalecited by 30,186
  4. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  5. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  6. 2017Graph Attention Networkscited by 8,923
  7. 2017Inductive Representation Learning on Large Graphscited by 6,320
  8. 2022Instant Neural Graphics Primitives with a Multiresolution Hash Encodingcited by 3,723
  9. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  10. 2019DeepSDF: Learning Continuous Signed Distance Functions for Shape Representationcited by 2,689
  11. 2019Fast Graph Representation Learning with PyTorch Geometriccited by 2,434
  12. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,333
27 more
  1. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  3. 2018Occupancy Networks: Learning 3D Reconstruction in Function Spacecited by 2,041
  4. 2018Learning Implicit Fields for Generative Shape Modelingcited by 1,060
  5. 2018On First-Order Meta-Learning Algorithmscited by 1,018
  6. 2020Learning Mesh-Based Simulation with Graph Networkscited by 868
  7. 2015Scheduled Sampling for Sequence Prediction with Recurrent Neural Networkscited by 803
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  9. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  10. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  11. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  12. 2022Message Passing Neural PDE Solverscited by 386
  13. 2017Meta-SGD: Learning to Learn Quickly for Few-Shot Learningcited by 379
  14. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
  15. 2017PDE-Net: Learning PDEs from Datacited by 311
  16. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  17. 2021Factorized Fourier Neural Operatorscited by 249
  18. 2020Learned Initializations for Optimizing Coordinate-Based Neural Representationscited by 242
  19. 2020MetaSDF: Meta-learning Signed Distance Functionscited by 214
  20. 2022From data to functa: Your data point is a function and you can treat it like onecited by 187
  21. 2021BACON: Band-limited Coordinate Networks for Multiscale Scene Representationcited by 143
  22. 2022Variable Bitrate Neural Fieldscited by 117
  23. 2017Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledgecited by 96
  24. 2022Clifford Neural Layers for PDE Modelingcited by 92
  25. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 73
  26. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 58
  27. 2022Physical Design using Differentiable Learned Simulatorscited by 37

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 24; 2025: 25; 2026: 33.

84 papers.

Sort:

By area:

  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  2. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  3. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 51
  4. 2024AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fieldscited by 39
  5. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 30
  6. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 28
  7. 2025Generative Latent Neural PDE Solver using Flow Matchingcited by 28
  8. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
  9. 2024Scale Equivariant Graph Metanetworkscited by 24
  10. 2024Aero-Nef: Neural Fields for Rapid Aircraft Aerodynamics Simulationscited by 24
  11. 2025Neural Chaos: A Spectral Stochastic Neural Operatorcited by 21
  12. 2024Operator Learning with Gaussian Processescited by 18
38 more
  1. 2024Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solvingcited by 17
  2. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  3. 2025Geometric Operator Learning with Optimal Transportcited by 16
  4. 2024CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equationscited by 14
  5. 2024Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equationscited by 12
  6. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  7. 2024Fast Training of Sinusoidal Neural Fields via Scaling Initializationcited by 11
  8. 2025Towards scalable surrogate models based on Neural Fields for large scale aerodynamic simulationscited by 10
  9. 2026Test-time Generalization for Physics through Neural Operator Splittingcited by 10
  10. 2025Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observationscited by 9
  11. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  12. 2026Structure-Preserving Learning Improves Geometry Generalization in Neural PDEscited by 8
  13. 2023Speeding up astrochemical reaction networks with autoencoders and neural ODEscited by 6
  14. 2025Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solvingcited by 6
  15. 2025Geometry aware inference of steady state PDEs using Equivariant Neural Fields representationscited by 6
  16. 2024Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformationscited by 5
  17. 2024Separable Operator Networkscited by 5
  18. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  19. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  20. 2025SCENT: Robust Spatiotemporal Learning for Continuous Scientific Data via Scalable Conditioned Neural Fieldscited by 4
  21. 2025PDEfuncta: Spectrally-Aware Neural Representation for PDE Solution Modelingcited by 4
  22. 2026SMART: Scalable Mesh-free Aerodynamic Simulations from Raw Geometries using a Transformer-based Surrogate Modelcited by 4
  23. 2024Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularizationcited by 3
  24. 2024Accelerating Simulation of Two-Phase Flows with Neural PDE Surrogatescited by 3
  25. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  26. 2026Topology-Preserving Neural Operator Learning via Hodge Decompositioncited by 3
  27. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  28. 2024A deformation-based framework for learning solution mappings of PDEs defined on varying domainscited by 2
  29. 2025Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solverscited by 2
  30. 2025Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modelingcited by 2
  31. 2025Conditional neural field for spatial dimension reduction of turbulence data: a comparison studycited by 2
  32. 2025Efficient Generative Transformer Operators For Million-Point PDEscited by 2
  33. 2026From Sparse Sensors to Continuous Fields: STRIDE for Spatiotemporal Reconstructioncited by 2
  34. 2026From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operatorscited by 2
  35. 2026Flowers: A Warp Drive for Neural PDE Solverscited by 2
  36. 2026Geometry-Aware Neural Optimizer for Shape Optimization and Inversioncited by 2
  37. 2026A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modelingcited by 2
  38. 2025Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operatorscited by 1

50 most cited of 84 All 84 (JSON)