Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Neural PDE solvers
Solves PDEs with convergence guarantees, coarse correction and latent spectral models.
135 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2018–2026) · an approach in Differential equations and PDEs, keyword cluster “pde · physics · partial differential”
State History Tried Baselines 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.
47 papers in 2023–24 and 64 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic shrank, from 26.4% to 21.4%.
Open questions the data points to
- Papers report PeRCNN on GS (RMSE) anywhere from 0.159 to 0.176: comparisons against it depend on whose number you take.
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
- 2018 · PDE-Net: Learning PDEs from Data (ICML 2018 · 206 citations)
- 2019 · Learning Neural PDE Solvers with Convergence Guarantees (ICLR 2019 · 51 citations)
- 2020 · Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction (ICML 2020 · 159 citations)
- 2021 · Low-Rank Registration Based Manifolds for Convection-Dominated PDEs (AAAI 2021 · 18 citations)
- 2022 · Message Passing Neural PDE Solvers (ICLR 2022 · 58 citations)
- 2023 · PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers (NeurIPS 2023 · 23 citations)
- 2024 · DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · 26 citations)
- 2025 · Interpretable Solutions for Multi-Physics PDEs Using T-NNGP (AAAI 2025 · 6 citations)
- 2026 · FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates (ICML 2026 · 0 citations)
Earliest papers here
- PDE-Net: Learning PDEs from Data (ICML 2018 · 206 citations)
- Learning Neural PDE Solvers with Convergence Guarantees (ICLR 2019 · 51 citations)
- Learning to Optimize Multigrid PDE Solvers (ICML 2019 · 29 citations)
What has been tried
104 papers here report 1,083 measured changes (a component added, removed or swapped, compared with a baseline): helped 71%, no clear effect 6%, hurt 19%, mixed 4%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| add PDE loss guidance to observation loss guidance | Helmholtz solution, sparse observations on coefficient and solution · relative error | 9.3% -> 0.6% | Helped | DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · 26 citations) |
| Reduce diversity by dropping trajectories for CE-CRP, CE-Gauss, and NS-Sines | downstream tasks · test error | Hurt | Poseidon: Efficient Foundation Models for PDEs (NeurIPS 2024 · 12 citations) | |
| set number of slices M to 1 instead of 64 | Elasticity · Relative L2 | 0.0064 -> 0.0148 | Hurt | Transolver: A Fast Transformer Solver for PDEs on General Geometries (ICML 2024 · 11 citations) |
| Add pruning operators | heat1 · average expression length | Helped | An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations (AAAI 2024 · 10 citations) | |
| add Newton loss to the MSE-only training objective | Convex PDE · test L2 error | Helped | Newton Informed Neural Operator for Solving Nonlinear Partial Differential Equations (NeurIPS 2024 · 9 citations) | |
| add 20-step unrolled training | 1D advection · rollout accuracy after 30 steps | Helped | APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs (NeurIPS 2024 · 8 citations) | |
| Add the decoupling strategy | Diffusion-convection · mean population fitness | Helped | Interpretable Solutions for Multi-Physics PDEs Using T-NNGP (AAAI 2025 · 6 citations) | |
| Use adaptive learnable diffusion instead of fixed-scale diffusion | CIFAR-10-C corruption distributions · accuracy | Helped | PDE+: Enhancing Generalization via PDE with Adaptive Distributional Diffusion (AAAI 2024 · 4 citations) | |
| Use latent evolution instead of input-space evolution | 2D Navier-Stokes forward, autoregressive rollout · MA | 0.0602 -> 0.0142 | Helped | Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution (AAAI 2024 · 3 citations) |
| Add self-modulation to FNO | Electromagnetics subdomain test · L_data | 8.04 -> 3.85; 6.11 -> 3.85 | Helped | Towards General Neural Surrogate Solvers with Specialized Neural Accelerators (ICML 2024 · 3 citations) |
| add symmetric convolution constraints | Burgers · RMSE | 0.1450 -> 0.0064 | Helped | P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics (NeurIPS 2024 · 3 citations) |
| Add LLM embeddings for equation symbols | 1D time-dependent PDEs, in-distribution · relative L2 | 0.0295 -> 0.0277; 5.76% | Helped | Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers (ICML 2025 · 2 citations) |
Baseline numbers that disagree
The same baseline on the same benchmark and metric, reported more than 5% apart by different papers here. Training budgets, data splits and reimplementations differ, so state which number you compare with.
| Baseline | Benchmark | Metric | Reported | Papers |
|---|---|---|---|---|
| PeRCNN | GS | RMSE | 0.159 – 0.176 | 2 papers, e.g. NeurIPS 2024 and ICML 2025 |
What reviewers will expect
Datasets and baselines named in the 50 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- Heat · 2 of 50 papers
- Combined equation · 2 of 50 papers
- PDEBench · 2 of 50 papers
- Plasticity · 2 of 50 papers
- Airfoil · 2 of 50 papers
- Darcy Flow · 2 of 50 papers
- Navier-Stokes · 2 of 50 papers
Baselines
- FNO · 17 of 50 papers
- U-Net · 7 of 50 papers
- PINN · 6 of 50 papers
- PINO · 5 of 50 papers
- MPP · 5 of 50 papers
- OFormer · 5 of 50 papers
- FactFormer · 5 of 50 papers
- F-FNO · 5 of 50 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 111 papers; from OpenAlex and the papers' affiliations).
- Microsoft · 8 papers
- Stanford University · 6 papers
- Tsinghua University · 5 papers
- Massachusetts Institute of Technology · 4 papers
- University of Amsterdam · 4 papers
- NEC · 4 papers
- Université Pierre et Marie Curie - Paris 6 · 4 papers
- Academy of Mathematics and Systems Science · 3 papers
- Chinese Academy of Sciences · 3 papers
- Westlake University · 3 papers
Use it
Download BibTeX of 37 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Physics-informed operator learning (129 papers)
- Neural operators for general geometries (111 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).