Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Neural operators for general geometries
Learns PDE solution operators on irregular domains using neural fields and latent operators.
111 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2020–2026) · an approach in Differential equations and PDEs, keyword cluster “pde · physics · partial differential”
State History Tried Checklist Preprints Who Export
Where it stands
Not enough benchmark data Fewer than four papers report the same benchmark, so there is no record to follow.
35 papers in 2023–24 and 66 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 19.7% to 22.1%.
How this is decided
A benchmark is tracked when at least four papers of this approach report their own method on the same dataset and metric in at least two different years. Its record is the best number reported up to 2024 compared with the best reported in 2025–26: still improving means at least 1% better, flat means at most 0.2% better with at least two papers from 2025–26 trying. Numbers on a different scale from the rest (a fraction among percentages) and far outliers are left out. Results come from different papers with different setups, so they are signals, not head-to-head comparisons. The share of papers is counted at ICML, NeurIPS, ICLR, CVPR and AAAI.
How it got here
Most cited paper of each year
- 2020 · Implicit Neural Representations with Periodic Activation Functions (NeurIPS 2020 · 262 citations)
- 2021 · Fourier Neural Operator for Parametric Partial Differential Equations (ICLR 2021 · 1,068 citations)
- 2022 · Solving PDE-Constrained Control Problems Using Operator Learning (AAAI 2022 · 16 citations)
- 2023 · Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations (AAAI 2023 · 21 citations)
- 2024 · Fourier Neural Operator with Learned Deformations for PDEs on General Geometries (NeurIPS 2024 · 150 citations)
- 2025 · Accurate Differential Operators for Hybrid Neural Fields (CVPR 2025 · 2 citations)
- 2026 · Cross-Field Interface-Aware Neural Operators for Multiphase Flow Simulation (AAAI 2026 · 1 citation)
Earliest papers here
- Implicit Neural Representations with Periodic Activation Functions (NeurIPS 2020 · 262 citations)
- Differential Equation Units: Learning Functional Forms of Activation Functions from Data (AAAI 2020 · 3 citations)
- Fourier Neural Operator for Parametric Partial Differential Equations (ICLR 2021 · 1,068 citations)
What has been tried
56 papers here report 615 measured changes (a component added, removed or swapped, compared with a baseline): helped 72%, no clear effect 6%, hurt 18%, mixed 5%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| Use learned deformation instead of fixed R-mesh deformation | Hyper-elasticity test · relative L2 error | 0.0536 -> 0.0229 | Helped | Fourier Neural Operator with Learned Deformations for PDEs on General Geometries (NeurIPS 2024 · 150 citations) |
| Replace PhCA with Physics-Attention | Forward PDE benchmarks · Relative L2 error | Darcy 0.49 -> 0.53; NS2d 8.45 -> 22.65; Airfoil 0.51 -> 0.74; Elasticity 0.52 -> 0.88; Plasticity 0.29 -> 0.49; Pipe 0.2 | Hurt | Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · 14 citations) |
| add local integral kernel branch to FNO | Darcy flow · relative L2-error | 6.034e-2 vs 5.867e-2 baseline | No clear effect | Neural Operators with Localized Integral and Differential Kernels (ICML 2024 · 10 citations) |
| use ASM instead of KLE for input dimension reduction | Hyperelasticity · relative L2 output reconstruction error | 3.8% -> 2.7% | Helped | Derivative-enhanced Deep Operator Network (NeurIPS 2024 · 5 citations) |
| Add orthogonal embedding to the MLP parameterization | Darcy; Airfoil; Pipe · l2 relative error | Darcy: 1.29e-2 -> 4.21e-3; Airfoil: 7.21e-3 -> 5.64e-3; Pipe: 7.84e-3 -> 3.44e-3 | Helped | Amortized Fourier Neural Operators (NeurIPS 2024 · 5 citations) |
| Increase pretraining samples from 2000 to 57545 | 2D incompressible Navier-Stokes · relative ℓ2 error | 0.3594 -> 0. 3246 | Helped | Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning (NeurIPS 2024 · 4 citations) |
| Add codomain attention | Fluid-structure interaction, NS+EW, 5 shots · L2 loss | 0.211 -> 0.040 | Helped | Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs (NeurIPS 2024 · 4 citations) |
| Use finite-difference supervision instead of polynomial-fit supervision | FamousShape · Ang | 3.19 -> 5.14 | Hurt | Accurate Differential Operators for Hybrid Neural Fields (CVPR 2025 · 2 citations) |
| Use insertion-based instead of appending-based construction | TSP1000 · Gap | 0.974% -> 0.481% | Helped | Learning to Insert for Constructive Neural Vehicle Routing Solver (NeurIPS 2025 · 2 citations) |
| train the NSM base architecture with PINN loss instead of spectral loss | Reaction-Diffusion · L2 relative error (%) | 0.20 ± 0.01 -> .075 ± .016; 0.48 ± 0.16 -> .086 ± .019; 0.78 ± 0.01 -> .083 ± .006; 1.28 ± 0.42 -> .077 ± .005 | Hurt | Neural Spectral Methods: Self-supervised learning in the spectral domain (ICLR 2024 · 2 citations) |
| Use STDE instead of parallelized backward mode SDGD | Two-body Allen-Cahn, 100K D · Speed (it/s) | 29.24 -> 156.90 | Helped | Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operators (NeurIPS 2024 · 2 citations) |
| replace Mamba integration with convolution integration | Navier-Stokes Equation · test errors | 1.00 × -> 1.14 × | Hurt | Alias-Free Mamba Neural Operator (NeurIPS 2024 · 2 citations) |
What reviewers will expect
Datasets and baselines named in the 36 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- Elasticity · 2 of 36 papers
Baselines
- FNO · 10 of 36 papers
- DeepONet · 4 of 36 papers
- Transolver · 4 of 36 papers
- F-FNO · 3 of 36 papers
- GNOT · 3 of 36 papers
- GINO · 3 of 36 papers
- UNET · 2 of 36 papers
- ResNet · 2 of 36 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 101 papers; from OpenAlex and the papers' affiliations).
- Tsinghua University · 5 papers
- California Institute of Technology · 5 papers
- ETHZ - ETH Zurich · 4 papers
- Swiss Federal Institute of Technology · 3 papers
- NVIDIA · 3 papers
- Amazon · 2 papers
- University of Maryland, College Park · 2 papers
- University of California, Berkeley · 2 papers
- Peking University · 2 papers
- University of Science and Technology of China · 2 papers
Use it
Download BibTeX of 34 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Neural PDE solvers (135 papers)
- Physics-informed operator learning (129 papers)
- Physics-informed neural network training (71 papers)
Elsewhere in the topic
- Neural ODE training and analysis (80 papers)
- Probabilistic inference for ODEs and SDEs (59 papers)
- Structured continuous-depth models (29 papers)
Benchmark numbers, ablations and baseline numbers are extracted by AI from each paper's full text and can be wrong; the paper is the authority. Data: results index of 2026-11-02, citation counts from OpenAlex (2026-10-09).