add readme

README.md

@@ -1,3 +1,271 @@
 ---
-license: unknown
+tags:
+- physics
+- simulation
+- FEM
+- PDE
+- neural-operator
+- scientific-computing
+- domain-decomposition
+size_categories:
+- 100K<n<1M
+pretty_name: SNI-Dataset
+dataset_creators:
+  - Bosch Center for Artificial Intelligence (BCAI)
+viewer: false
 ---
+
+# SNI-Data
+
+<p align="center">
+  <img src=".huggingface/Bosch_logo.png" alt="Bosch Logo" width="200">
+</p>
+
+<p align="center">
+  <em>Created by <a href="https://www.bosch-ai.com/">Bosch Center for Artificial Intelligence (BCAI)</a></em>
+  <br>
+  <strong>Paper:</strong> <a href="https://arxiv.org/abs/2504.00510">Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving</a> (ICLR 2026)
+</p>
+
+A benchmark of 2D finite-element PDE solutions for training and evaluating neural operators with geometry generalization. Each sample is a complete FEM problem defined on an unstructured triangular mesh—random polygon geometry, boundary conditions (Dirichlet and/or Neumann), and optionally coefficient fields or time-stepping parameters—paired with the solved solution field $u$.
+
+The dataset accompanies the **Schwarz Neural Inference (SNI)** framework, which combines local operator learning with domain decomposition methods to generalize to unseen complex geometries at inference time.
+
+## Dataset Summary
+
+| Property | Value |
+|---|---|
+| Domain | 2D Partial Differential Equations (FEM) |
+| Number of PDE types | 5 |
+| Training samples | 200,000 |
+| Test samples (simple domains) | 26,500 |
+| Test samples (evaluation domains) | 1,330 |
+| Total samples | ~227,830 |
+| File format | Pickle (`.pkl`) |
+| Mesh type | Unstructured triangular (gmsh) |
+| Solver | FEniCSx (dolfinx) |
+
+## PDE Types
+
+| PDE | Equation | Boundary Conditions | Type |
+|---|---|---|---|
+| `laplace2d` | $-\nabla^2 u = 0$ | Dirichlet | Stationary |
+| `laplace2d_mixed` | $-\nabla^2 u = 0$ | Mixed Dirichlet / Neumann | Stationary |
+| `darcy2d` | $-\nabla \cdot (a(x)\nabla u) = f(x)$ | Dirichlet | Stationary |
+| `heat2d` | $\partial u / \partial t - \alpha \nabla^2 u = 0$ | Time-dependent Dirichlet | Transient |
+| `nonlinear_poisson2d` | $-\nabla \cdot (q(u)\nabla u) = 0$, $q(u) = 1 + u^2$ | Dirichlet | Stationary (nonlinear) |
+
+### PDE Details
+
+**Laplace (Dirichlet):** The classical Laplace equation on random polygonal domains with randomized Dirichlet boundary values at each boundary node.
+
+**Laplace (Mixed):** Same equation, but the boundary is split into contiguous Dirichlet and Neumann segments. With probability 0.2 the entire boundary is Dirichlet; otherwise a random contiguous portion is assigned Neumann conditions.
+
+**Darcy Flow:** A variable-coefficient elliptic PDE. The coefficient field $a(x)$ and source term $f(x)$ are independently randomized per node ($a \in [0, 1]$, $f \in [-5, 0]$). Boundary values are scaled by a random factor in $[0.3, 1.0]$.
+
+**Heat Equation:** Time-dependent diffusion solved with implicit Euler. Training uses 10 time steps ($T = 0.1$, $\Delta t = 0.01$) with random thermal diffusivity $\alpha \in [0.1, 1.0]$. Evaluation uses 50 time steps ($T = 0.5$) with fixed $\alpha = 1.0$.
+
+**Nonlinear Poisson:** A nonlinear PDE with solution-dependent diffusivity $q(u) = 1 + u^2$, solved via Newton's method with GMRES and BoomerAMG preconditioning.
+
+## Training Data
+
+Training data is generated on **random simple polygons** with varying numbers of vertices, triangulated using gmsh.
+
+| Subset | Samples | Polygons &times; Batch | Vertices | Mesh Size |
+|---|---|---|---|---|
+| `laplace2d_simple` | 20,000 | 250 &times; 10 | 3–12 | 0.1 |
+| `laplace2d_mixed_simple` | 40,000 | 10 &times; 20 | 3–12 | 0.1 |
+| `darcy2d_simple` | 40,000 | 250 &times; 10 | 3–16 | 0.1 |
+| `heat2d_simple` | 80,000 | 160 &times; 50 | 3–12 | 0.1 |
+| `nonlinear_poisson2d_simple` | 20,000 | 250 &times; 10 | 3–12 | 0.1 |
+
+> **Note:** Each random polygon is reused for multiple samples (the "Batch" count) with different boundary conditions and/or coefficient fields. Coordinates are shifted by $[0.5, 0.5]$ to center domains around the origin.
+
+## Test Data
+
+### Simple Domain Test Sets
+
+Test data on random polygons (same generation process as training, different samples):
+
+| Subset | Samples |
+|---|---|
+| `laplace2d_simple` | 2,500 |
+| `laplace2d_mixed_simple` | 4,000 |
+| `darcy2d_simple` | 2,500 |
+| `heat2d_simple` | 12,500 |
+| `nonlinear_poisson2d_simple` | 2,500 |
+
+### Evaluation Domain Test Sets
+
+Test data on **pre-defined complex geometries** (fixed meshes), used to evaluate geometry generalization:
+
+| Domain | Mesh File | Description |
+|---|---|---|
+| **A** (Schwarz) | `A-schwarz.msh` | Overlapping disk and rectangle |
+| **B** (Holes) | `B-holes.msh` | Square with two interior holes |
+| **C** (Bosch) | `C-bosch.msh` | Disk with complex shape removed |
+
+Each PDE is evaluated on each domain with 100 samples (10 for `heat2d`):
+
+| Subset | Domain A | Domain B | Domain C |
+|---|---|---|---|
+| `laplace2d` | 100 | 100 | 100 |
+| `laplace2d_mixed` | 100 | 100 | 100 |
+| `darcy2d` | 100 | 100 | 100 |
+| `heat2d` | 10 | 10 | 10 |
+| `nonlinear_poisson2d` | 100 | 100 | 100 |
+
+Additional evaluation meshes (`D-dolphin.msh`, `E-disk.msh`, `F-triangle.msh`) are available for extended evaluation.
+
+## Data Format
+
+All data is stored in Python pickle files (`.pkl`). Each file contains a **list of samples**. The format varies by PDE type:
+
+### Laplace2d / Laplace2d Mixed / Nonlinear Poisson2d
+
+Each sample is a tuple: `(sol, [bc])`
+
+| Array | Shape | Description |
+|---|---|---|
+| `sol` | `(N, 3)` | Solution field: `[x, y, u]` at each mesh node |
+| `bc` | `(M, 4)` | Boundary conditions: `[x, y, value, type]` at each boundary node |
+
+- `type = 0` → Dirichlet boundary condition ($u = \text{value}$)
+- `type = 1` → Neumann boundary condition ($\partial u / \partial n = \text{value}$)
+- For `laplace2d` and `nonlinear_poisson2d`, all boundaries are Dirichlet (`type = 0`)
+
+### Darcy2d
+
+Each sample is a tuple: `(sol, [qf, bc])`
+
+| Array | Shape | Description |
+|---|---|---|
+| `sol` | `(N, 3)` | Solution field: `[x, y, u]` |
+| `qf` | `(N, 4)` | Coefficient and source: `[x, y, a, f]` at each mesh node |
+| `bc` | `(M, 4)` | Boundary conditions: `[x, y, u_D, 0]` |
+
+### Heat2d
+
+Each sample is a tuple: `(sol, alpha, [bc])`
+
+| Array | Shape | Description |
+|---|---|---|
+| `sol` | `(N, 2 + T)` | Solution trajectory: `[x, y, u_0, u_1, ..., u_{T-1}]` |
+| `alpha` | scalar | Thermal diffusivity |
+| `bc` | `(M, 2 + T + 1)` | Boundary trajectory: `[x, y, bc_0, ..., bc_{T-1}, 0]` |
+
+- Training: $T = 10$ time steps ($\Delta t = 0.01$)
+- Evaluation: $T = 50$ time steps ($\Delta t = 0.01$)
+
+## Directory Structure
+
+```
+data/
+├── 2d/
+│   ├── laplace2d_simple_20000_train.pkl
+│   ├── laplace2d_simple_2500_test.pkl
+│   ├── laplace2d_schwarz_100_test.pkl
+│   ├── laplace2d_holes_100_test.pkl
+│   ├── laplace2d_bosch_100_test.pkl
+│   ├── laplace2d_mixed_simple_40000_train.pkl
+│   ├── laplace2d_mixed_schwarz_100_test.pkl
+│   ├── laplace2d_mixed_holes_100_test.pkl
+│   ├── laplace2d_mixed_bosch_100_test.pkl
+│   ├── darcy2d_simple_40000_train.pkl
+│   ├── darcy2d_simple_2500_test.pkl
+│   ├── darcy2d_schwarz_100_test.pkl
+│   ├── darcy2d_holes_100_test.pkl
+│   ├── darcy2d_bosch_100_test.pkl
+│   ├── heat2d_simple_100000_train.pkl
+│   ├── heat2d_simple_12500_test.pkl
+│   ├── heat2d_schwarz_10_test.pkl
+│   ├── heat2d_holes_10_test.pkl
+│   ├── heat2d_bosch_10_test.pkl
+│   ├── nonlinear_poisson2d_simple_20000_train.pkl
+│   ├── nonlinear_poisson2d_simple_2500_test.pkl
+│   ├── nonlinear_poisson2d_schwarz_100_test.pkl
+│   ├── nonlinear_poisson2d_holes_100_test.pkl
+└── └── nonlinear_poisson2d_bosch_100_test.pkl
+```
+
+## Quick Start
+
+```python
+import pickle
+
+# Load training data
+with open("data/2d/laplace2d_simple_20000_train.pkl", "rb") as f:
+    datalist = pickle.load(f)
+
+# Each sample is a tuple
+sol, (bc,) = datalist[0]
+
+# Solution field
+x, y, u = sol[:, 0], sol[:, 1], sol[:, 2]  # (N,) each
+
+# Boundary conditions
+bx, by, bc_val, bc_type = bc[:, 0], bc[:, 1], bc[:, 2], bc[:, 3]
+
+print(f"Number of samples: {len(datalist)}")
+print(f"Mesh nodes: {sol.shape[0]}, Boundary nodes: {bc.shape[0]}")
+```
+
+```python
+# Load Darcy flow data (includes coefficient field)
+with open("data/2d/darcy2d_simple_40000_train.pkl", "rb") as f:
+    datalist = pickle.load(f)
+
+sol, (qf, bc) = datalist[0]
+
+# Coefficient field a(x) and source f(x)
+a_coeff = qf[:, 2]  # (N,)
+f_source = qf[:, 3]  # (N,)
+```
+
+```python
+# Load Heat equation data (time-dependent)
+with open("data/2d/heat2d_simple_100000_train.pkl", "rb") as f:
+    datalist = pickle.load(f)
+
+sol, alpha, (bc,) = datalist[0]
+
+# Solution at each time step
+x, y = sol[:, 0], sol[:, 1]
+u_timesteps = sol[:, 2:]  # (N, 10) — solution at 10 time steps
+print(f"Thermal diffusivity: {alpha}")
+```
+
+## Data Generation
+
+Data is generated using [FEniCSx](https://fenicsproject.org/) (dolfinx) for the FEM solver and [gmsh](https://gmsh.info/) for mesh generation:
+
+```bash
+# Generate training data (parallelized across processes)
+bash scripts/generate_data.sh laplace2d train 8
+
+# Generate evaluation data on pre-defined domains
+python data_generation/generate_eval.py --pde all --domain all
+```
+
+See the [project repository](https://arxiv.org/abs/2504.00510) for the full training and inference pipeline.
+
+## Intended Use
+
+SNI-Dataset is designed to:
+- Train and benchmark **neural operators** for PDE solving on irregular geometries.
+- Evaluate **geometry generalization** — training on simple random polygons, testing on complex unseen domains.
+- Support research on **domain decomposition methods** combined with learned operators.
+- Provide a diverse set of PDE types (elliptic, parabolic, nonlinear) with varying boundary condition types.
+
+## Citation
+
+If you use SNI-Dataset in your work, please cite:
+
+```bibtex
+@inproceedings{
+  title={Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2026},
+  url={https://arxiv.org/abs/2504.00510}
+}
+```