---
pretty_name: TopoBox-3D
language:
  - en
tags:
  - neural-operators
  - partial-differential-equations
  - scientific-machine-learning
  - topology
  - hodge-laplacian
---

# TopoBox-3D

TopoBox-3D is the dataset accompanying **Beyond Arbitrary Geometry: Topology
Generalization in Neural PDE Operators**. It is a controlled three-dimensional
benchmark for separating fixed-topology geometry shift from generalization to
unseen homological support.

The benchmark contains 5,280 connected box-minus-void geometries and 63,360
fixed-time Hodge-heat instances. Through-tunnels and enclosed cavities control
the first and second Betti numbers. Every geometry is represented by a
tetrahedral mesh, geometry features, a regular-grid signed-distance field, and
an oriented simplicial complex. Hodge-heat data are provided for vertex, edge,
and face cochains (`k = 0, 1, 2`) under four initial-condition configurations.

## Scope

| Item | Count |
|---|---:|
| Protocols | 4 |
| Geometries per protocol | 1,320 |
| Geometries in total | 5,280 |
| Degrees per geometry | 3 |
| Initial conditions per degree | 4 |
| PDE instances in total | 63,360 |
| Geometry HDF5 shards | 108 |
| Hodge-heat HDF5 shards | 212 |

Each protocol has 800 training, 120 validation, 200 Test-IID, and 200 Test-OOD
geometries. Geometry IDs are the atomic split unit.

| Protocol | In-support topology | Test-OOD topology | Shift |
|---|---|---|---|
| A | `(beta1, beta2) = (1, 1)`, family A | `(1, 1)`, family B | fixed-topology geometry |
| B | `beta1 in {0,1,2}, beta2 = 0` | `(3, 0)` | unseen tunnel support |
| C | `beta1 = 0, beta2 in {0,1,2}` | `(0, 3)` | unseen cavity support |
| D | `(beta1, beta2) in {0,1,2}^2` | `(3, 3)` | mixed topology |

## Directory layout

```text
TopoBox-3D/
├── DATASET.md                 detailed geometry schema
├── dataset_config.json        generation and protocol configuration
├── manifest.csv               one row per geometry
├── packed/                    training-ready geometry HDF5 shards
│   ├── index.csv
│   ├── index.json
│   └── protocol_{A,B,C,D}/...
└── protocol_{A,B,C,D}/...     raw per-geometry mesh data

TopoBox-3D-HodgeHeat/
├── manifest.json              equation and generation configuration
├── index.csv
├── index.json                 geometry-to-shard lookup
├── COMPLETION.json            completion and adapter checks
├── audit_report.json          deep numerical audit
└── protocol_{A,B,C,D}/...     Hodge-heat HDF5 shards

examples/
└── TopoBox-3D-HodgeHeat-representatives/
                                lightweight topology and field previews

RELEASE.json                   release-level counts and provenance
SHA256SUMS.txt                 checksums for all published files
```

The raw geometry layer contains `mesh.npz`, `mesh.msh`, `mesh.vtu`, and
`metadata.json` for every geometry. The packed layer stores the same numerical
content in HDF5 shards optimized for training. Both layers are included so the
release supports efficient experiments, per-sample inspection, and independent
repacking. The small `TopoBox-3D-mini` development subset is not duplicated in
this repository because it is derived from the complete release. The
`examples/` directory contains only lightweight previews referenced by the
saved completion record; it is not an additional data split.

## Hodge-heat task

The supervised target is the fixed-time solution of

```text
partial_t omega + kappa Delta_k omega = 0,
k in {0,1,2}, kappa = 1, T = 0.1.
```

Targets use homogeneous absolute boundary conditions and 100
Crank--Nicolson steps. The four initial-condition configurations are
`non_harmonic`, `weak_harmonic`, `balanced`, and `strong_harmonic`. Geometry and
PDE records are joined by `geometry_id`.

## Loading with the accompanying code

After placing this dataset under the code repository's `data/` directory, the
expected roots are:

```text
data/TopoBox-3D/packed/
data/TopoBox-3D-HodgeHeat/
```

```python
from topobox3d.pde_dataset import TopoBoxPDEDataset

dataset = TopoBoxPDEDataset(
    geometry_packed_root="data/TopoBox-3D/packed",
    solution_root="data/TopoBox-3D-HodgeHeat",
    protocol="B",
    split="train",
    degrees=(1,),
    configs=("balanced",),
)

sample = dataset[0]
print(sample.geometry_id, sample.w0.shape, sample.wT.shape)
dataset.close()
```

The accompanying code repository contains the generators, validators, model
adapters, training entry points, and complete schema documentation. Public
paper and code links will be added when the anonymous review period permits.

## Integrity and validation

The geometry manifest and both HDF5 indices contain 5,280 unique geometry IDs.
The Hodge-heat release contains 212 shards and 63,360 PDE instances. The saved
deep audit reports zero errors. `SHA256SUMS.txt` can be used to verify the local
copy after download.

## License and citation

License and final citation metadata have intentionally not been asserted during
anonymous review. They must be added before the public dataset is released.