Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Physics-informed operator learning
Embeds physical constraints in DeepONets and diffusion models for fluid and graph PDEs.
129 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 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.
32 papers in 2023–24 and 78 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 18.0% to 26.1%.
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
- 2018 · Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge (ICLR 2018 · 76 citations)
- 2020 · Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods (AAAI 2020 · 213 citations)
- 2021 · Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting (ICLR 2021 · 113 citations)
- 2022 · PDEBench: An Extensive Benchmark for Scientific Machine Learning (NeurIPS 2022 · 56 citations)
- 2023 · Self-Consistent Velocity Matching of Probability Flows (NeurIPS 2023 · 5 citations)
- 2024 · Neural Oscillators for Generalization of Physics-Informed Machine Learning (AAAI 2024 · 19 citations)
- 2025 · Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints (NeurIPS 2025 · 4 citations)
- 2026 · AdaField: Generalizable Surface Pressure Modeling with Physics-Informed Pre-training and Flow-Conditioned Adaptation (AAAI 2026 · 3 citations)
Earliest papers here
- Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge (ICLR 2018 · 76 citations)
- Variational PDEs for Acceleration on Manifolds and Application to Diffeomorphisms (NeurIPS 2018 · 7 citations)
- Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods (AAAI 2020 · 213 citations)
What has been tried
69 papers here report 612 measured changes (a component added, removed or swapped, compared with a baseline): helped 68%, no clear effect 7%, hurt 21%, 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 |
|---|---|---|---|---|
| Use lower oscillator Δt | Viscous Burgers equation · error | Helped | Neural Oscillators for Generalization of Physics-Informed Machine Learning (AAAI 2024 · 19 citations) | |
| add constrained local flux collocation points alongside IC and mass constraints | Burgers IC · generation quality | Helped | Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints (NeurIPS 2025 · 4 citations) | |
| Train for longer, up to 50000 epochs | Value function estimation · Prediction Error (%) | 7.104 -> 5.322 -> 5.475 -> 5.940 -> 4.733 | Helped | Physics-Informed Representation and Learning: Control and Risk Quantification (AAAI 2024 · 4 citations) |
| use pure data-driven integration instead of numerical integration | Burgers · RMSE_u (cm/s) | 0.0074 -> 0.0236 | Hurt | PARCv2: Physics-aware Recurrent Convolutional Neural Networks for Spatiotemporal Dynamics Modeling (ICML 2024 · 3 citations) |
| Infer the initial anatomy instead of fixing it to an average brain | Recurrence (Any) · Recurrence Coverage [%] | 73.4 ± 3.2 -> 74.7 ± 3.1 | Helped | Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization (NeurIPS 2024 · 3 citations) |
| Allow unconstrained lambda values and non-monotonic inner functions rather than enforcing positive lambda and increasing inner functions | PINN experiments · performance | No clear effect | Deep Learning Alternatives Of The Kolmogorov Superposition Theorem (ICLR 2025 · 2 citations) | |
| Increase features per layer from 4 to 32 | Compressible 1D SPH · test error | by an order of magnitude | Helped | Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics (ICLR 2024 · 2 citations) |
| Add the advection ODE form | ERA5 forecasting · unweighted RMSE | Helped | ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs (ICLR 2024 · 2 citations) | |
| add diffusion flow (DF) | Burgers2d C Int. and C Ext. · mean L2 relative error ε | 0.067 -> 0.039; 0.207 -> 0.101 | Helped | PAPM: A Physics-aware Proxy Model for Process Systems (ICML 2024 · 2 citations) |
| add narrow-field-of-view (small interval) optimization | PSF accuracy, lens #63762, no noise, H=1 · PSNR/SSIM | 41.064/ 0.922 -> 49.252/0.981 | Helped | A Physics-Informed Blur Learning Framework for Imaging Systems (CVPR 2025 · 1 citation) |
| fine-tune with PDE-Aligner rather than use pre-training alone | 1D PDEs · nRMSE | 0.0103 -> 0.0056 | Helped | OmniArch: Building Foundation Model for Scientific Computing (ICML 2025 · 1 citation) |
| Optimize all modes simultaneously instead of adding them through boosting | Training tasks · accuracy | Helped | Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems (ICML 2025 · 1 citation) |
What reviewers will expect
Datasets and baselines named in the 42 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- PDEBench · 2 of 42 papers
Baselines
- PINNs · 2 of 42 papers
- FNO · 2 of 42 papers
- MLP · 2 of 42 papers
- SIREN · 2 of 42 papers
- KAN · 2 of 42 papers
- PINN · 2 of 42 papers
- Adam · 2 of 42 papers
- DeepONet · 2 of 42 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 110 papers; from OpenAlex and the papers' affiliations).
- Technical University Munich · 4 papers
- Technische Universität München · 4 papers
- Massachusetts Institute of Technology · 4 papers
- National University of Singapore · 4 papers
- Stanford University · 3 papers
- NVIDIA · 3 papers
- Peking University · 3 papers
- Imperial College London · 3 papers
- Technical University of Munich · 2 papers
- Tsinghua University · 2 papers
Use it
Download BibTeX of 36 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Neural PDE solvers (135 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).