Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Structured continuous-depth models
Build continuous-depth and differential-equation-based networks with conservation, sparsity, or higher-order structure.
29 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2018–2026) · an approach in Differential equations and PDEs, keyword cluster “ordinary · odes · differential equations”
State History Tried 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.
8 papers in 2023–24 and 3 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic shrank, from 4.5% to 1.0%.
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 · Learning Across Scales---Multiscale Methods for Convolution Neural Networks (AAAI 2018 · 63 citations)
- 2019 · AReS and MaRS Adversarial and MMD-Minimizing Regression for SDEs (ICML 2019 · 11 citations)
- 2020 · Deep Learning For Symbolic Mathematics (ICLR 2020 · 46 citations)
- 2021 · On the Validity of Modeling SGD with Stochastic Differential Equations (SDEs) (NeurIPS 2021 · 15 citations)
- 2022 · Learning Optimal K-Space Acquisition and Reconstruction Using Physics-Informed Neural Networks (CVPR 2022 · 22 citations)
- 2023 · Neural signature kernels as infinite-width-depth-limits of controlled ResNets (ICML 2023 · 2 citations)
- 2024 · AdaFlow: Imitation Learning with Variance-Adaptive Flow-Based Policies (NeurIPS 2024 · 7 citations)
- 2025 · BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization (NeurIPS 2025 · 0 citations)
- 2026 · Score Based Error Correcting Code Decoder (ICML 2026 · 0 citations)
Earliest papers here
- Learning Across Scales---Multiscale Methods for Convolution Neural Networks (AAAI 2018 · 63 citations)
- AReS and MaRS Adversarial and MMD-Minimizing Regression for SDEs (ICML 2019 · 11 citations)
- A Solvable High-Dimensional Model of GAN (NeurIPS 2019 · 10 citations)
What has been tried
25 papers here report 194 measured changes (a component added, removed or swapped, compared with a baseline): helped 69%, no clear effect 9%, 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 variance estimation network to AdaFlow | Maze 1 · success rate | 0.78 -> 0.98 | Helped | AdaFlow: Imitation Learning with Variance-Adaptive Flow-Based Policies (NeurIPS 2024 · 7 citations) |
| Set initial-distribution support size D to 1 instead of D=2 | CATS-Regions Normal, m=50 · revenue | 589.50 -> 278.82 | Hurt | BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization (NeurIPS 2025 · 0 citations) |
| Increase attention depth from h=2 to h=3 or h=4 | Spring-mass, k-link pendulum, and elastic pendulum · MSE loss | 2.58e-6 -> 2.49e-6; 4.26e-9 -> 4.39e-9; 5.41e-7 -> 5.28e-7 (h=3); 2.58e-6 -> 2.55e-6; 4.26e-9 -> 4.42e-9; 5.41e-7 -> 5.5 | Mixed | AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios (ICML 2024 · 0 citations) |
| Set alternating frequency γ to 1 | MNAR simulation · convergence stability | Hurt | DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation (ICML 2024 · 0 citations) |
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 11 papers; from OpenAlex and the papers' affiliations).
- Imperial College London · 2 papers
- Shanghai Jiao Tong University · 2 papers
Use it
Download BibTeX of 26 key papersOpen the topic in the interactive map
Get alerts. Describe your idea in scime atlas and save it (free scime account): you get a weekly email when new preprints land close to it.
Related approaches
Same keyword cluster
- Neural ODE training and analysis (80 papers)
- Probabilistic inference for ODEs and SDEs (59 papers)
Elsewhere in the topic
- Neural PDE solvers (135 papers)
- Physics-informed operator learning (129 papers)
- Neural operators for general geometries (111 papers)
- Physics-informed neural network training (71 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).