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

# TopoBox-3D

[Paper (arXiv:2609.05860)](https://arxiv.org/abs/2609.05860) | [Code (GitHub)](https://github.com/asmld/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](https://github.com/asmld/TopoBox-3D)
contains the generators, validators, model adapters, training entry points,
and complete schema documentation. See the
[paper](https://arxiv.org/abs/2609.05860) for the benchmark and reported results.

## 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

Dataset license metadata has not yet been specified.

For the paper, see [Beyond Arbitrary Geometry: Topology Generalization in Neural PDE Operators (arXiv:2609.05860)](https://arxiv.org/abs/2609.05860).