---
license: mit
task_categories:
  - other
tags:
  - pde
  - optimal-control
  - operator-learning
  - deeponet
  - differentiable-predictive-control
  - scientific-machine-learning
pretty_name: PDE Control with DPC and Time-Integrated Neural Operators
size_categories:
  - 1K<n<10K
---

# PDE Control — TI-DeepONet Training Datasets

Simulation datasets for **"Learning to Control PDEs with Differentiable Predictive Control
and Time-Integrated Neural Operators"**.

- 📄 Paper: [arXiv:2511.08992](https://arxiv.org/abs/2511.08992)
- 💻 Code: [github.com/Centrum-IntelliPhysics/PDEControl_DPC](https://github.com/Centrum-IntelliPhysics/PDEControl_DPC)

These are the trajectory datasets used to train the **TI-DeepONet surrogates**. Each file
holds 3000 controlled trajectories generated by a classical numerical solver under randomly
sampled control sequences.

> **You may not need these.** The pretrained checkpoints are committed in the code
> repository, and both notebooks in each experiment directory run without any download.
> You need these files only to retrain a surrogate from scratch.

## Files

| File | System | Size | Trajectories | Timesteps | Controls |
|---|---|---|---|---|---|
| `heat_dataset_dpc.npz` | 1D heat equation | 895 MB | 3000 | 401 | 4 |
| `burgers_smooth_f_dataset.npz` | 1D Burgers' equation | 670 MB | 3000 | 301 | 2 |
| `reaction_diffusion_dataset200.npz` | 1D reaction–diffusion | 893 MB | 3000 | 401 | 4 |

All arrays are `float64`. Spatial resolution is $N = 100$ on $x \in [0,1]$ with
$\Delta t = 10^{-3}$ throughout.

## Contents

Each `.npz` contains:

| Key | Shape | Meaning |
|---|---|---|
| `solutions` | `(3000, T+1, 100)` | state trajectories $u(x,t)$ |
| `controls` | `(3000, T, n_c)` | control amplitudes $c_i(t)$ |
| `x` | `(100,)` | spatial grid, `linspace(0, 1, 100)` |
| `dt` | scalar | time step, `1e-3` |

Plus per-system physical parameters:

- **heat** — `nu` (diffusivity, 0.1), `centers` (4 actuator positions), `sigma` (actuator width)
- **burgers** — no extra keys; actuator geometry lives in the code's `config.py`
- **reaction–diffusion** — `D` (diffusivity, 0.01), `r` (reaction rate, 1.0), `centers`, `sigma`

Control enters every system as a sum of Gaussian sources:

$$f(x,t) = \sum_{i=1}^{n_c} c_i(t)\,\exp\!\left(-\frac{(x - x_i)^2}{2\sigma^2}\right)$$

## Generation

| System | Solver |
|---|---|
| Heat | Crank–Nicolson |
| Burgers' | Upwind advection + forward Euler |
| Reaction–diffusion | Backward Euler + Newton iteration, Neumann BCs |

The generating scripts are in the code repository (`heat_1D_gen.py`,
`Burgers_1D_smooth_f_gen.py`, `RD_1D_gen2.py`), so every file here is reproducible from
source.

## Usage

```bash
git clone https://github.com/Centrum-IntelliPhysics/PDEControl_DPC.git
cd PDEControl_DPC
pip install -r requirements.txt

python download_data.py            # all three
python download_data.py heat       # or just one
```

Files land where each `config.py` expects them. Then:

```bash
cd HE_TT && python train_ti_don.py --epochs 120000
```

To load one directly:

```python
import numpy as np
d = np.load("heat_dataset_dpc.npz")
print(d["solutions"].shape)   # (3000, 401, 100)
print(d["controls"].shape)    # (3000, 400, 4)
```

## Citation

```bibtex
@misc{sarkar2025learningcontrolpdesdifferentiable,
      title={Learning to Control PDEs with Differentiable Predictive Control and Time-Integrated Neural Operators},
      author={Dibakar Roy Sarkar and Ján Drgoňa and Somdatta Goswami},
      year={2025},
      eprint={2511.08992},
      archivePrefix={arXiv},
      primaryClass={cs.CE},
      url={https://arxiv.org/abs/2511.08992},
}
```

## License

MIT, matching the code repository.