Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Probabilistic inference for ODEs and SDEs
Estimate parameters and solutions of differential equations with probabilistic solvers and state-space inference.
59 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2016–2026) · an approach in Differential equations and PDEs, keyword cluster “ordinary · odes · differential equations”
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.
16 papers in 2023–24 and 21 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic shrank, from 9.0% to 7.0%.
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
- 2016 · Probabilistic Linear Multistep Methods (NeurIPS 2016 · 13 citations)
- 2018 · Algorithmic Linearly Constrained Gaussian Processes (NeurIPS 2018 · 18 citations)
- 2019 · GOODE: A Gaussian Off-The-Shelf Ordinary Differential Equation Solver (ICML 2019 · 7 citations)
- 2020 · ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systems (AAAI 2020 · 19 citations)
- 2021 · Learning Stochastic Behaviour from Aggregate Data (ICML 2021 · 6 citations)
- 2022 · DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps (NeurIPS 2022 · 284 citations)
- 2023 · ODE-RSSM: Learning Stochastic Recurrent State Space Model from Irregularly Sampled Data (AAAI 2023 · 10 citations)
- 2024 · Constant Acceleration Flow (NeurIPS 2024 · 3 citations)
- 2025 · Discovering Symmetries of ODEs by Symbolic Regression (AAAI 2025 · 2 citations)
- 2026 · Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling (ICML 2026 · 0 citations)
Earliest papers here
- Probabilistic Linear Multistep Methods (NeurIPS 2016 · 13 citations)
- Fast Parameter Inference in Nonlinear Dynamical Systems using Iterative Gradient Matching (ICML 2016 · 6 citations)
- Algorithmic Linearly Constrained Gaussian Processes (NeurIPS 2018 · 18 citations)
What has been tried
50 papers here report 430 measured changes (a component added, removed or swapped, compared with a baseline): helped 69%, no clear effect 7%, hurt 20%, 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 reflow to 1-Rectified flow | CIFAR-10, N=1 · FID | 378 -> 6.88 | Helped | Constant Acceleration Flow (NeurIPS 2024 · 3 citations) |
| Replace the basic EDM coefficient adjustment with coefficients exploiting the internal structure of the EDM update | CIFAR10, FFHQ, AFHQV2, and ImageNet64 at fewer than 25 NFEs · FID | Helped | On Accelerating Diffusion-Based Sampling Processes via Improved Integration Approximation (ICLR 2024 · 2 citations) | |
| Remove the categorical sampling step | text8, fewer than 50 NFE · spelling accuracy (SA) | Helped | Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations (ICML 2024 · 2 citations) | |
| Use early stopping during tempering except at the final diffusion value | HH₁, 2 parameters · ITER | 696.22 ± 78.10 -> 197.81 ± 40.38 | Helped | Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations (ICML 2024 · 2 citations) |
| Add EDL to NSDE without WBPR | MovieLens-100K · P@5 | 0.4065 -> 0.4523 | Helped | Evidential Stochastic Differential Equations for Time-Aware Sequential Recommendation (NeurIPS 2024 · 2 citations) |
| add structural evolution to the graph (rather than keep it fixed) | Social, prediction length 10 · MAPE | 11.61 -> 9.15 | Helped | Pioneer: Physics-informed Riemannian Graph ODE for Entropy-increasing Dynamics (AAAI 2025 · 1 citation) |
| use HBF with discretization error O(η³) instead of HBF with O(η²) | 2-d model · discretization error | Helped | Heavy-Ball Momentum Method in Continuous Time and Discretization Error Analysis (NeurIPS 2025 · 1 citation) | |
| increase Lanczos budget from 50 MVs to 150 MVs | Bayesian VAN calibration · Marginal likelihood (log) | -192,757 -> -154,475 | Helped | Gradients of Functions of Large Matrices (NeurIPS 2024 · 1 citation) |
| Use state-dependent instead of state-independent volatility | Stochastic Lotka–Volterra system · trajectory structure | Helped | SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations (ICML 2025 · 0 citations) | |
| Add RSCO constant optimization | PDE benchmarks · complete recovery rate | Helped | Closed-form Solutions: A New Perspective on Solving Differential Equations (ICML 2025 · 0 citations) | |
| Use Eulerian self-distillation, which requires spatial derivatives | Image generation, including CIFAR-10 · training stability | Hurt | How to build a consistency model: Learning flow maps via self-distillation (NeurIPS 2025 · 0 citations) | |
| Increase n_eta from 1 to 2 | Wave 1D · normalized error | 0.517 -> 0.210 | Helped | Learning Stochastic Multiscale Models (NeurIPS 2025 · 0 citations) |
What reviewers will expect
Datasets and baselines named in the 15 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- CIFAR-10 · 2 of 15 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 37 papers; from OpenAlex and the papers' affiliations).
- Tsinghua University · 3 papers
- University of Tuebingen · 3 papers
- Kobe University · 2 papers
- University of Tübingen · 2 papers
- Toronto University · 2 papers
- Korea Advanced Institute of Science and Technology · 2 papers
Use it
Download BibTeX of 37 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Neural ODE training and analysis (80 papers)
- Structured continuous-depth models (29 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).