PartialObs–PDEBench

What is this website?

PartialObs–PDEBench focuses on PDE reconstruction and inference when observations are sparse (missing sensors, masked pixels, partial trajectories).

Manuscript scope: the repository supports one data/benchmark paper—dataset design, task/split/metric protocols, anchor baselines, difficulty analysis, and one-line tools. New semantic-ID, large world-model, and foundation-model claims are separate projects.

Research: browse/search 301 AI4PDE/AI4SDE papers (12 curated pages + index placeholders).
PDE problems: which PDEs appear in the literature + which papers use them.
Baselines: a cross-paper index of commonly compared methods.
Benchmark: machine-checked 7×4×10 design, seven task protocols, seven split views, anchor matrix, per-sample analysis records, and publication-candidate quality gates.
Run: copy-ready Linux server and SeaWulf Slurm examples that start from this Git repository.
Contribute: how to add/curate papers via JSON.

Paper tree

A compact, conceptual lineage map (selected famous works).

ASCII (clickable)

AI4PDE / SciML (selected milestones)
├─ Physics-informed optimization
│  ├─ Deep Ritz (2018)
│  ├─ DGM / Deep Galerkin (2018)
│  ├─ DeepBSDE (2018)
│  └─ PINNs (2019)
│     ├─ cPINNs (2020)
│     ├─ SA-PINNs (2020)
│     ├─ XPINNs (2021)
│     ├─ gPINNs (2021)
│     └─ FBPINNs (2021)
├─ Operator learning
│  ├─ FNO (2020)
│  ├─ GKN (2020)
│  ├─ MGNO (2020)
│  ├─ DeepONet (2021)
│  ├─ PINO (2021)
│  ├─ Galerkin Transformer (2021)
│  ├─ U-NO (2022)
│  ├─ WNO (2022)
│  ├─ CNO (2023)
│  └─ U-WNO (2024)
├─ Diffusion / generative PDE inference
│  ├─ Conditional diffusion protocols (2024)
│  ├─ DiffusionPDE (2024)
│  ├─ FunDPS (2025)
│  ├─ PRISMA (2025)
│  └─ VideoPDE (2025)
├─ Graph / mesh simulators
│  ├─ GNS (ICML 2020)
│  └─ MeshGraphNets (ICLR 2021)
└─ Benchmarks and datasets
   ├─ PDEBench (2022)
   ├─ PDEArena (2022)
   ├─ FourCastNet (2022)
   └─ GraphCast (2023)

Mermaid (clickable)

flowchart TD
  Root["AI4PDE / SciML (selected milestones)"]

  %% Physics-informed optimization (PINN family)
  Root --> PI["Physics-informed optimization"]
  PI --> DeepRitz["Deep Ritz (2018)"]
  PI --> DGM["DGM / Deep Galerkin (2018)"]
  PI --> DeepBSDE["DeepBSDE (2018)"]
  PI --> PINN["PINNs (2019)"]
  PINN --> cPINN["cPINNs (2020)"]
  PINN --> SAPINN["SA-PINNs (2020)"]
  PINN --> XPINN["XPINNs (2021)"]
  PINN --> gPINN["gPINNs (2021)"]
  PINN --> FBPINN["FBPINNs (2021)"]

  %% Operator learning (neural operators)
  Root --> OL["Operator learning"]
  OL --> DeepONet["DeepONet (2021)"]
  OL --> FNO["FNO (2020)"]
  FNO --> PINO["PINO (2021)"]
  FNO --> GalerkinT["Galerkin Transformer (2021)"]
  FNO --> UNO["U-NO (2022)"]
  FNO --> WNO["WNO (2022)"]
  WNO --> UWNO["U-WNO (2024)"]
  FNO --> CNO["CNO (2023)"]
  OL --> GKN["GKN (2020)"]
  OL --> MGNO["MGNO (2020)"]

  %% Diffusion / generative inference
  Root --> DiffGen["Diffusion / generative PDE inference"]
  DiffGen --> CondDiff["Conditional diffusion protocols (2024)"]
  CondDiff --> DiffPDE["DiffusionPDE (2024)"]
  DiffPDE --> FunDPS["FunDPS (2025)"]
  FunDPS --> PRISMA["PRISMA (2025)"]
  DiffPDE --> VideoPDE["VideoPDE (2025)"]

  %% Graph simulators
  Root --> GraphSim["Graph / mesh simulators"]
  GraphSim --> GNS["GNS (ICML 2020)"]
  GraphSim --> MGN["MeshGraphNets (ICLR 2021)"]

  %% Benchmarks / datasets
  Root --> Bench["Benchmarks and datasets"]
  Bench --> PDEBench["PDEBench (2022)"]
  Bench --> PDEArena["PDEArena (2022)"]
  Bench --> FourCastNet["FourCastNet (2022)"]
  FourCastNet --> GraphCast["GraphCast (2023)"]

  %% Clickable links (homepage)
  %% - Paper nodes go to curated pages.
  %% - Category nodes go to the Research tab with an initial filter.
  click Root "research/" "Open the research index" _self

  click PI "research/?method=PINN%20%2F%20physics-constrained" "Filter: PINN / physics-constrained" _self
  click DeepRitz "research/paper/?slug=deep-ritz" "Deep Ritz (2018)" _self
  click DGM "research/paper/?slug=dgm" "Deep Galerkin Method (2018)" _self
  click DeepBSDE "research/paper/?slug=deepbsde" "DeepBSDE (2018)" _self
  click PINN "research/paper/?slug=pinn" "PINNs (2019)" _self
  click cPINN "research/paper/?slug=cpinn" "cPINNs (2020)" _self
  click SAPINN "research/paper/?slug=sa-pinn" "SA-PINNs (2020)" _self
  click XPINN "research/paper/?slug=xpinn" "XPINNs (2021)" _self
  click gPINN "research/paper/?slug=gpinn" "gPINNs (2021)" _self
  click FBPINN "research/paper/?slug=fbpinns" "FBPINNs (2021)" _self

  click OL "research/?method=Operator%20learning" "Filter: Operator learning" _self
  click DeepONet "research/paper/?slug=deeponet" "DeepONet (2021)" _self
  click FNO "research/paper/?slug=fno" "Fourier Neural Operator (2020)" _self
  click PINO "research/paper/?slug=pino" "Physics-Informed Neural Operator (2021)" _self
  click GalerkinT "research/paper/?slug=galerkin-transformer" "Galerkin Transformer (2021)" _self
  click UNO "research/paper/?slug=u-no" "U-NO (2022)" _self
  click WNO "research/paper/?slug=wno" "WNO (2022)" _self
  click UWNO "research/paper/?slug=u-wno" "U-WNO (2024)" _self
  click CNO "research/paper/?slug=cno" "CNO (2023)" _self
  click GKN "research/paper/?slug=gkn" "Graph Kernel Network (2020)" _self
  click MGNO "research/paper/?slug=mgno" "MGNO (2020)" _self

  click DiffGen "research/?method=Diffusion" "Filter: Diffusion" _self
  click DiffPDE "research/paper/?slug=diffusionpde" "DiffusionPDE (2024)" _self
  click FunDPS "research/paper/?slug=fundps" "FunDPS (2025)" _self
  click PRISMA "research/paper/?slug=prisma" "PRISMA (2025)" _self
  click VideoPDE "research/paper/?slug=videopde" "VideoPDE (2025)" _self

  click GraphSim "research/?method=Graph%20%2F%20mesh" "Filter: Graph / mesh" _self
  click GNS "research/paper/?slug=gns" "GNS (ICML 2020)" _self
  click MGN "research/paper/?slug=meshgraphnets" "MeshGraphNets (ICLR 2021)" _self

  click Bench "benchmark/" "Benchmark tab" _self
  click PDEBench "research/paper/?slug=pdebench" "PDEBench (2022)" _self
  click PDEArena "research/paper/?slug=pdearena" "PDEArena (2022)" _self
  click FourCastNet "research/paper/?slug=fourcastnet" "FourCastNet (2022)" _self
  click GraphCast "research/paper/?slug=graphcast" "GraphCast (2023)" _self

  click CondDiff "research/paper/?slug=conditional-diffusion-pde" "Open paper page"

  %% Theme tweaks
  classDef cat fill:#e9eeff,stroke:#3555d1,color:#0f172a;
  classDef node fill:#ffffff,stroke:#c9d4e4,color:#0f172a;
  class Root,PI,OL,DiffGen,GraphSim,Bench cat;
  class DeepRitz,DGM,DeepBSDE,PINN,cPINN,SAPINN,XPINN,gPINN,FBPINN,DeepONet,FNO,PINO,GalerkinT,UNO,WNO,UWNO,CNO,GKN,MGNO,CondDiff,DiffPDE,FunDPS,PRISMA,VideoPDE,GNS,MGN,PDEBench,PDEArena,FourCastNet,GraphCast node;

AI4PDE + AI4SDE map

A high-level taxonomy you can extend as new method families emerge.

flowchart TD
  R["AI4PDE + AI4SDE: a taxonomy (high-level)"]

  R --> Phys["Physics-constrained learning"]
  Phys --> PINNfam["PINN-style residual minimization"]
  Phys --> Hybrid["Hybrid: data + physics losses"]

  R --> Op["Operator learning"]
  Op --> NO["Neural operators (FNO/DeepONet/...)"]
  Op --> ROM["Learned ROM / reduced models"]

  R --> Graph["Graph / mesh simulators"]
  Graph --> MP["Message passing / GNN solvers"]
  Graph --> Mesh["Mesh-based neural fields"]

  R --> Gen["Generative / probabilistic modeling"]
  Gen --> Score["Score-based / diffusion models"]
  Gen --> Bridge["Diffusion/SDE bridges (conditioning)"]
  Gen --> UQ["Uncertainty quantification"]

  R --> Theory["Theory & guarantees"]
  Theory --> Approx["Approximation / expressivity"]
  Theory --> Stability["Stability / generalization"]

  R --> Bench["Benchmarks"]

  %% Clickable links (homepage)
  click R "research/" "Open the research index" _self
  click Phys "research/?method=PINN%20%2F%20physics-constrained" "Filter: PINN / physics-constrained" _self
  click PINNfam "research/?method=PINN%20%2F%20physics-constrained" "Filter: PINN / physics-constrained" _self
  click Hybrid "research/?q=hybrid" "Search: hybrid" _self

  click Op "research/?method=Operator%20learning" "Filter: Operator learning" _self
  click NO "research/?q=neural%20operator" "Search: neural operator" _self
  click ROM "research/?q=reduced%20order" "Search: reduced order" _self

  click Graph "research/?method=Graph%20%2F%20mesh" "Filter: Graph / mesh" _self
  click MP "research/?q=message%20passing" "Search: message passing" _self
  click Mesh "research/?q=mesh" "Search: mesh" _self

  click Gen "research/?method=Diffusion" "Filter: Diffusion" _self
  click Score "research/?method=Diffusion" "Filter: Diffusion" _self
  click Bridge "research/?q=bridge" "Search: bridge" _self
  click UQ "research/?q=uncertainty" "Search: uncertainty" _self

  click Theory "research/?q=theory" "Search: theory" _self
  click Approx "research/?q=approximation" "Search: approximation" _self
  click Stability "research/?q=stability" "Search: stability" _self

  click Bench "benchmark/" "Benchmark tab" _self

Show ASCII fallback

AI4PDE + AI4SDE (taxonomy)
├─ Physics-informed optimization (PINN family)
├─ Operator learning (neural operators)
├─ Graph / mesh simulators
├─ Generative inference (diffusion / SDE bridges)
└─ Benchmarks and datasets