PINNeAPPle-Labs/darcy-2d-li2021

2-D Darcy flow, Li et al. (2021) operator-learning benchmark (1200 coefficient / pressure pairs, 128 x 128)

-div(a grad u) = 1 on the unit square, u = 0 on the boundary. Coefficients a = 12 where g >= 0 and 3 where g < 0, g ~ N(0, (-Laplacian + 9 I)^-2) (Li et al. 2021, section 5.2). g is drawn as a periodic Gaussian random field on the 128 x 128 cell grid (Li et al. use a cosine expansion on 421 nodes), so a is piecewise constant on the cells of the stored grid. The task maps a to u. Cell centres ((i + 1/2) / 128, (j + 1/2) / 128); u is the cell average of the solution.

darcy_2d_li2021 v1.0.0 (revision v1.0.0.post1) · licence cc-by-4.0 · part of the PINNeAPPle zoo

Configs and splits

Config	Split	Samples	Parquet rows	Parquet files	Content
default	train	1000	1000	1	1000 pairs
default	test	200	200	1	200 pairs (held out)

Split rule: rows 0-999 are train, 1000-1199 test; inputs are i.i.d. draws, so the contiguous split is a random split

Columns

Column	Unit	Type	Description
a	-	float32[128, 128]	permeability on the 128 x 128 cells (values 3 and 12)
index	-	int32	sample index (row of the generated set)
u	-	float32[128, 128]	pressure, cell averages on the 128 x 128 cells

Load

With the library (the npz files, SHA-256 checked; arrays with their full shape):

pip install "pinneapple @ git+https://github.com/PINNeAPPle-Labs/PINNeAPPle"

import pinneapple as pp

d = pp.datasets.load("darcy_2d_li2021", split="test")
print(d.n_rows, d.columns)
print(d["a"].shape)

With datasets (the Parquet files of this repo):

from datasets import load_dataset

ds = load_dataset("PINNeAPPle-Labs/darcy-2d-li2021", split="test", revision="v1.0.0.post1")
print(ds)
ds = ds.with_format("numpy")
print(ds[0]["a"].shape)

Provenance

python scripts/zoo/operator_data.py generate --out DIR (inputs drawn from fixed seeds in the parent process, so the data do not depend on the number of workers); solvers in pinneapple_simulation.numerical_solvers.operator_benchmarks, written from the equations; seed 20212

Verification

Solver (cell-centred two-point-flux finite volumes, harmonic face coefficients, sparse direct solve) checked in tests/test_operator_benchmarks.py on a manufactured variable-coefficient solution (second order, max error < 2e-3 at 64^2). Each sample is solved on the 512^2 grid (the piecewise-constant coefficient is represented exactly on every refinement) and averaged onto its 128^2 cells. Grid study on 4 held-aside inputs against 1024^2: max rel. L2 error 4.6e-3 (128^2), 1.6e-3 (256^2), 4.3e-4 (512^2, the dataset); observed order 1.36 / 1.21 (reduced from 2 by the coefficient jumps); Richardson estimate of the dataset's discretisation error ~8e-4 rel. L2, ten times below the ~1e-2 operator-learning methods reach on this task. u > 0 and below the a = 3 bound 1/24 in every sample.

Files

The npz files are the library's format and the source of truth: pp.datasets.load downloads them from this repo at the tagged revision and refuses any file whose SHA-256 differs from the manifest of the package. The Parquet files under data/ are derived from them by pinneapple_hub.zoo.parquet.write_dataset and hold the same numbers (a test round-trips npz -> Parquet -> arrays bit for bit); they are what the dataset viewer and load_dataset read.

File	Size	SHA-256
train.npz	59.2 MB	e32d8114482e9bcef831ab291938bbe0fd9306a24fb675d95a5735a98d1cc72d
test.npz	11.8 MB	c7f7b49deb5eff530405f44a8b2f8246d8c666b898b32ebeb6a18e82b049e89e

Total 71.1 MB of npz.

Citation

Li Z., Kovachki N., Azizzadenesheli K., Liu B., Bhattacharya K., Stuart A., Anandkumar A. (2021). Fourier Neural Operator for Parametric Partial Differential Equations. ICLR 2021, arXiv:2010.08895. Data generated with PINNeAPPle's reference solvers (pinneapple_simulation.numerical_solvers.operator_benchmarks); PINNeAPPle, Zenodo DOI 10.5281/zenodo.23343062.

Cite

If you use this dataset, please cite PINNeAPPle (software, Zenodo DOI 10.5281/zenodo.23343062; see CITATION.cff):

@software{pinneapple,
  author = {Barros, Yan and Schiassi, Enrico},
  title  = {PINNeAPPle},
  doi    = {10.5281/zenodo.23343062},
  url    = {https://github.com/PINNeAPPle-Labs/PINNeAPPle}
}