mlsanityData to 14 Oct 2026

Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations

arXiv 2406.03919 ↗First posted 6 Jun 2024cs.LG cs.AI cs.CVcited by 12

Not on the map. Why?

Abstract (arXiv)

Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed, all of which suffer from the typical problems of Transformers, such as quadratic memory and time complexity. Furthermore, all prevalent architectures for PDE solving lack at least one of several desirable properties of an ideal surrogate model, such as (i) generalization to PDE parameters not seen during training, (ii) spatial and temporal zero-shot super-resolution, (iii) continuous temporal extrapolation, (iv) support for 1D, 2D, and 3D PDEs, and (v) efficient inference for longer temporal rollouts. To address these limitations, we propose Vectorized Conditional Neural Fields (VCNeFs), which represent the solution of time-dependent PDEs as neural fields. Contrary to prior methods, however, VCNeFs compute, for a set of multiple spatio-temporal query points, their solutions in parallel and model their dependencies through attention mechanisms. Moreover, VCNeF can condition the neural field on both the initial conditions and the parameters of the PDEs. An extensive set of experiments demonstrates that VCNeFs are competitive with and often outperform existing ML-based surrogate models.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Builds on 34 of its 46 references that are in the map. First posted on arXiv 6 Jun 2024.

Builds on

34 of 46 references.

References · 34

  1. 2017Attention Is All You Needcited by 60,361
  2. 2020An Image is Worth 16x16 Words: Transformers for Image Recognition at Scalecited by 30,186
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,333
  5. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  6. 2018Occupancy Networks: Learning 3D Reconstruction in Function Spacecited by 2,041
  7. 2020Transformers are RNNs: Fast Autoregressive Transformers with Linear Attentioncited by 2,022
  8. 2020Conformer: Convolution-augmented Transformer for Speech Recognitioncited by 1,863
  9. 2020Learning to Simulate Complex Physics with Graph Networkscited by 880
  10. 2021CrossViT: Cross-Attention Multi-Scale Vision Transformer for Image Classificationcited by 806
  11. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  12. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
22 more
  1. 2021Neural Fields in Visual Computing and Beyondcited by 540
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  3. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  4. 2022Message Passing Neural PDE Solverscited by 386
  5. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  6. 2021Choose a Transformer: Fourier or Galerkincited by 343
  7. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
  8. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  9. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 216
  10. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  11. 2023Scalable Transformer for PDE Surrogate Modelingcited by 128
  12. 2020Transformers for Modeling Physical Systemscited by 106
  13. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 98
  14. 2021Learnable Fourier Features for Multi-Dimensional Spatial Positional Encodingcited by 93
  15. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 84
  16. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 73
  17. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 58
  18. 2022Physics Informed Neural Fields for Smoke Reconstruction with Sparse Datacited by 52
  19. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 38
  20. 2022Implicit Neural Spatial Representations for Time-dependent PDEscited by 35
  21. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  22. 2022Attention Beats Concatenation for Conditioning Neural Fieldscited by 20

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 5; 2026: 4.

12 papers.

  1. 2024Active Learning for Neural PDE Solverscited by 22
  2. 2024Deep vectorised operators for pulsatile hemodynamics estimation in coronary arteries from a steady-state priorcited by 6
  3. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  4. 2024Physics-informed reduced order model with conditional neural fieldscited by 4
  5. 2026SMART: Scalable Mesh-free Aerodynamic Simulations from Raw Geometries using a Transformer-based Surrogate Modelcited by 4
  6. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  7. 2025Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learningcited by 2
  8. 2025Multimodal Neural Operators for Real-Time Biomechanical Modelling of Traumatic Brain Injurycited by 1
  9. 2025Generalizable Implicit Neural Representations via Parameterized Latent Dynamics for Baroclinic Ocean Forecastingcited by 0
  10. 2026AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modelingcited by 0
  11. 2026A Deep Learning-based surrogate model for Severe Accidents in nuclear reactors using ASTECcited by 0
  12. 2026PDE-JEPA: Predictive Representation Learning of Latent Dynamics Modeling for Parametric PDEscited by 0