PDE-OBS: physical records

Complete numerical solutions for the seven PDE families of the PDE-OBS benchmark, generated by the solvers in the benchmark's own code. Nothing was downloaded, scraped or converted from another corpus, and the same specification reproduces the same record. The trained baseline checkpoints are in the companion model repository ru1ch3n/PDE-OBS.

What is here

Axis	Value
PDE families	Darcy, Poisson, Helmholtz (stationary); Heat, reaction–diffusion, Burgers, Navier–Stokes (temporal)
Boundary protocols	Dirichlet, Neumann, periodic, Robin with obstacle
Condition-field constructions	smooth_grf, medium_grf, rough_grf, low_frequency_fourier, multi_frequency_fourier, gaussian_blobs, piecewise_blocks, threshold_level_set, front_ring_shock, dipole_vortex_pair
Records	2,000 per (family, boundary, construction), divided 667 / 667 / 666 across the three physical regimes; 280 such combinations in 840 regime nodes, 560,000 records in total
Resolution	128 × 128; 1 stored frame for stationary families, 15 equally spaced states for temporal families
Format	HDF5 shards, channel-last, float32, gzip level 4. A regime node's 667 or 666 records are written as 200 / 200 / 200 / 67 or 66, so the fourth shard of every node is a short one
Size	3,360 shards, 244 GB, plus per-shard sidecars

The paper-evaluated slice is the subset under data/<family>/<paper boundary>/smooth_grf/ (Dirichlet for the stationary families, periodic for the temporal ones): 14,000 records in 84 shards, 6.5 GB with sidecars. Everything else is the rest of the factorial design the benchmark supports.

Each shard stores condition [N,H,W,1], trajectory [N,T,H,W,1], geometry [N,H,W,1] and one JSON metadata string per record carrying its identity, family, boundary, construction, regime, seeds and stored times. Beside every shard: .manifest.json (schema, record count, byte size, SHA-256), .sha256, .quality.json (the generation quality gates), and .metadata.json / .metadata.csv exports of the per-record metadata.

Observation masks are not stored here. They are constructed at access time from the record identity and the protocol name, which is what lets one physical record serve every observation pattern. The benchmark code defines the nine frozen views and their exact observed-cell counts.

How to use it

Two checksum manifests are published in this repository. Both follow the benchmark's release-manifest v1 contract (schema version 1, one entry per file with its SHA-256, byte size and URL) and both list the sidecars the loader's verify mode checks against.

Manifest	Contents	Size
release_manifest.json	the paper-evaluated slice, 84 shards + 252 sidecars	6.5 GB
release_manifest_full.json	the complete corpus, 3,360 shards + 10,080 sidecars	244 GB

With the benchmark's downloader, which validates the manifest before fetching, resumes interrupted files and verifies every file's SHA-256 on arrival:

pdeobs download --tier full --output ./pdeobs-data \
    --manifest https://huggingface.co/datasets/ru1ch3n/PDE-OBS/resolve/main/release_manifest.json

Then build a task instance:

from pdeobs import api

data = api.load_dataset("./pdeobs-data", verify=True)      # checks every shard against its sidecars
obs = api.make_observation("paper:R50")                    # one of the nine frozen views
package, targets = api.inference_input_from_dataset(data, obs, task="recovery")

Or fetch the raw files without the benchmark:

from huggingface_hub import snapshot_download
root = snapshot_download("ru1ch3n/PDE-OBS", repo_type="dataset",
                         allow_patterns=["data/poisson/dirichlet/smooth_grf/*"])

Provenance and limits

• Generated by the benchmark's deterministic solvers with seed 20260804; elliptic solves use relative tolerance 1e-9 and raise rather than returning a partial solution.
• Every record passed the generation quality gates: all stored values finite, geometry binary error ≤ 1e-6, normalized initial-condition / boundary-condition / initial-transition defects ≤ 1e-6 / 1e-4 / 5e-6, and a normalized PDE residual ≤ 0.05.
• These are discrete-residual checks. No independent reference-solution or grid-convergence study is included, so the records are not certified against an external numerical reference.
• The Dirichlet families impose zero values on the outer stored layer of a cell-centered grid; that boundary placement is part of the numerical specification and differs from an endpoint-coordinate convention.
• Parameter values are unitless numbers on the unit domain; no SI units are assigned.
• Every published shard's SHA-256 equals its .sha256 sidecar and its entry in the two release manifests; all 3,360 were re-hashed before upload and every file was verified on the Hub after upload.

De-identification

The shards were re-emitted before publication with the generation provenance de-identified: the output path, interpreter path, host name and scheduler block inside each shard's spec_json attribute, the per-record git_commit, and the same fields in every sidecar were replaced by placeholders. The numerical datasets (condition, trajectory, geometry) were copied chunk by chunk without recompression and are bit-identical to the generated corpus. Because the attributes changed, the published shard digests differ from the generation-time ones; scrub-manifest.json lists the rewritten files with their published SHA-256 (the correspondence to the generation-time digests is kept with the benchmark code, beside the results). summary.json and summary.quality.json / summary.quality.csv are the generation-time corpus summary and quality tables with the same fields de-identified, the data-root label reduced to numerics-full-t15, and their shard digests re-pointed to the published files.

October 2026 revision data (october2026/)

The V2 data of the October 2026 revision: the views the models in october2026/ of the companion model repository were trained and scored on, and the independent confirmation set. Generated by the benchmark's own solvers (pdeobs.revision, data contract pdeobs.rev8k.v2.data/1); 36.4 GB in 6,372 shards.

october2026/v2/<view>/<regime>/<split>/part_NNNNN.h5          training / validation / test views
october2026/v2-confirm/<view>/<regime>/<split>/part_NNNNN.h5  confirmation set (fresh parents, scored once)
    beside every shard: .manifest.json  .sha256  .quality.json  .metadata.json  .metadata.csv
october2026/data_manifest.part-<n>.json   per upload part: per view files, bytes, shards, records, digest;
                                          per shard path, bytes, SHA-256
october2026/results/part-A/, part-B/      run-level (part A) and per-record (parts A and B) test errors

<view> is v2_<pde>_<ell>_ref128[__grid<N>][__t4]. ell is the length scale of the Gaussian-random-field construction (ell0025 rough, ell009 medium, ell022 smooth); each slice has the regimes low / medium / high with 3,000 training, 300 validation and 600 test parents each. __grid<N> is the same parents on a 64^2 or 256^2 grid, not new samples; __t4 is the Navier-Stokes forecast view (vorticity u0 and frames 1-3). The confirmation set has 200 further parents per regime and was sealed until every model, view list and setting was frozen.

Directory	View	Shards	Records	GB
october2026/v2/	v2_darcy_ell0025_ref128	468	11,700	1.19
october2026/v2/	v2_darcy_ell009_ref128	468	11,700	1.15
october2026/v2/	v2_darcy_ell022_ref128	468	11,700	1.12
october2026/v2/	v2_ns_ell0025_ref128__grid256__t4	468	11,700	11.97
october2026/v2/	v2_ns_ell0025_ref128__grid64__t4	468	11,700	0.88
october2026/v2/	v2_ns_ell0025_ref128__t4	468	11,700	3.16
october2026/v2/	v2_ns_ell009_ref128__t4	468	11,700	3.02
october2026/v2/	v2_ns_ell022_ref128__t4	468	11,700	2.84
october2026/v2/	v2_poisson_ell0025_ref128	468	11,700	1.27
october2026/v2/	v2_poisson_ell0025_ref128__grid256	468	11,700	4.71
october2026/v2/	v2_poisson_ell0025_ref128__grid64	468	11,700	0.37
october2026/v2/	v2_poisson_ell009_ref128	468	11,700	1.22
october2026/v2/	v2_poisson_ell022_ref128	468	11,700	1.19
october2026/v2-confirm/	v2_darcy_ell0025_ref128	24	600	0.06
october2026/v2-confirm/	v2_darcy_ell009_ref128	24	600	0.06
october2026/v2-confirm/	v2_darcy_ell022_ref128	24	600	0.06
october2026/v2-confirm/	v2_ns_ell0025_ref128	24	600	0.50
october2026/v2-confirm/	v2_ns_ell0025_ref128__t4	24	600	0.16
october2026/v2-confirm/	v2_ns_ell009_ref128	24	600	0.48
october2026/v2-confirm/	v2_ns_ell009_ref128__t4	24	600	0.15
october2026/v2-confirm/	v2_ns_ell022_ref128	24	600	0.45
october2026/v2-confirm/	v2_ns_ell022_ref128__t4	24	600	0.15
october2026/v2-confirm/	v2_poisson_ell0025_ref128	24	600	0.06
october2026/v2-confirm/	v2_poisson_ell009_ref128	24	600	0.06
october2026/v2-confirm/	v2_poisson_ell022_ref128	24	600	0.06

<download>/october2026/v2 is the --data-root of reproduce/october/replay_inference.py in the benchmark code. The observation layouts (U256, C256, L256, MXR3, ...) are not stored; they are built at access time by pdeobs.revision.v2masks from the parent identity and the protocol name.

Results. october2026/results/part-A/ holds runs_long.csv.gz, classical_long.csv.gz and per_record/<suite>.csv.gz: the exported files whose SHA-256 are listed in results/revision_october/manifest.part-A.json of the benchmark code (a byte-identical copy of the manifest is beside them). Per-record rows are run_id, view, sample_id, error[, regime, obs, hid, hid_rel]. october2026/results/part-B/ holds part B's classical_long.csv.gz and per_record/<suite>.csv.gz (SHA-256 as in manifest.part-B.json, copied beside them). Part B's run-level export runs_long.csv.gz is not published here, because two voided-attempt rows carry a machine path in their error text; its run-level table is the committed results/revision_october/runs_long.part-B.csv. The later exports are published beside them, earlier files unchanged: october2026/results/part-A/update_20261021/ (the 2026-10-21 export of part A: the same suite files with the E6 / E6ev rows of 2026-10-20 and the N7 confirmation rows of the P4 U1024 specialists added; 11 files, export_manifest.json lists their SHA-256) and october2026/results/part-B/update_20261021/ (the 2026-10-21 export of part B with the E5 composition-reference rows; 7 files, again without the run-level export). The update_20261020 / update_20261021 blocks of the committed manifests describe the added rows.

• Unchanged files. The shards and sidecars are published exactly as generated: every file was scanned before upload (shard attributes, string datasets, every sidecar) and carries no machine path, account, host or scheduler identifier, so no rewriting was needed. Each shard's SHA-256 equals its .sha256 sidecar and its entry in the data manifest; after the upload every file was checked on the Hub without credentials.
• Not included: the RPB view (third-party data, see the model card); the generation receipts (job bookkeeping of the generating cluster, which carries scheduler identifiers).

Citation

Ruichen Xu, Siyao Wang, Fang Wan, Jiacheng Qiu, Wenhan Gao, Jiaxing Zhang, Linsey Pang, Ravid Shwartz-Ziv, Yann LeCun and Yuefan Deng. PDE-OBS: Controlled Evaluation Across Observation Patterns. arXiv preprint arXiv:2609.36521, 2026.

@article{xu2026pdeobs,
  title   = {{PDE-OBS}: Controlled Evaluation Across Observation Patterns},
  author  = {Xu, Ruichen and Wang, Siyao and Wan, Fang and Qiu, Jiacheng and Gao, Wenhan and Zhang, Jiaxing and
             Pang, Linsey and Shwartz-Ziv, Ravid and LeCun, Yann and Deng, Yuefan},
  journal = {arXiv preprint arXiv:2609.36521},
  year    = {2026}
}