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[ICLR 2025] Wavelet Diffusion Neural Operator (WDNO) uses diffusion models on wavelet space for generative PDE simulation and control.

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# WDNO: Wavelet Diffusion Neural Operator (ICLR 2025)

[arXiv](https://arxiv.org/abs/2412.04833) ｜ [Paper](https://openreview.net/forum?id=FQhDIGuaJ4)

Official repo for the paper [Wavelet Diffusion Neural Operator](https://arxiv.org/abs/2412.04833).<br />
[Peiyan Hu*](https://peiyannn.github.io/), [Rui Wang*](https://scholar.google.ca/citations?hl=zh-CN&user=8VTaeFwAAAAJ), [Xiang Zheng](https://openreview.net/profile?id=~Xiang_Zheng5), [Tao Zhang](https://zhangtao167.github.io), [Haodong Feng](https://scholar.google.com/citations?user=0GOKl_gAAAAJ&hl=en), [Ruiqi Feng](https://weenming.github.io/), [Long Wei](https://longweizju.github.io/), [Yue Wang](https://www.microsoft.com/en-us/research/people/yuwang5/), [Zhi-Ming Ma](http://homepage.amss.ac.cn/research/homePage/8eb59241e2e74d828fb84eec0efadba5/myHomePage.html), [Tailin Wu†](https://tailin.org/).<br />
ICLR 2025.

We introduce Wavelet Diffusion Neural Operator (WDNO), a novel method for generative PDE simulation and control, to address diffusion models' challenges of modeling system states with abrupt changes and generalizing to higher resolutions.

Framework of WDNO:

## News
We develop a WDNO that shares a unified input–output interface with standard neural operators in the [RealPDEBench repository](https://github.com/AI4Science-WestlakeU/RealPDEBench).

## Environment
Run the following commands to install dependencies. In particular, when running the 2D control task, the Python version must be 3.8 due to the requirement of the Phiflow software.

```code
bash env.sh
```

## Datasets
The datasets can be downloaded respectively in [link](https://drive.google.com/drive/folders/1W1tbQ7ltIDEQdHzUarFo9EYMe4ngxXcz).
Please place the corresponding datasets in the `data` folder under the experiment directory.

## Checkpoints
The checkpoints can be downloaded respectively in [link](https://drive.google.com/drive/u/2/folders/1qjYXG53Y6cSK24EeyKLqab3kAzs961we).
Please place the corresponding checkpoints in the `results` folder under the experiment directory.

## 1D Burgers' Equation Simulation
Prepare data for WDNO:
```code
cd burgers
python wave_trans.py
```
Training of Base-Resolution Model:
```code
bash /scripts/burgers/train_base_sim.sh
```
Inference of Base-Resolution Model:
```code
bash /scripts/burgers/eval_base_sim.sh
```
Training of Super-Resolution Model:
```code
bash /scripts/burgers/train_super_sim.sh
```
Inference of Super-Resolution Model:
```code
bash /scripts/burgers/eval_super_sim.sh
```

## 2D Smoke Simulation
Prepare data for WDNO:
```code
cd smoke
python wave_trans_2d.py
```
Training of Base-Resolution Model:
```code
bash /scripts/smoke/train_base_sim.sh
```
Inference of base-resolution:
```code
bash /scripts/smoke/inf_base_sim.sh
```
Training of Super-Resolution Model:
```code
bash /scripts/smoke/train_super_sim.sh
```
Inference of super-resolution:
```code
bash /scripts/smoke/inf_super_sim.sh
```

## 1D Burgers' Equation Control:
Prepare data for WDNO:
```code
cd burgers
python wave_trans.py
```
Training:
```code
bash /scripts/burgers/train_base_control.sh
```
Inference:
```code
bash /scripts/burgers/eval_base_control.sh
```

## 2D Smoke Control:
Prepare data for WDNO:
```code
cd smoke
python wave_trans_2d.py
```
training:
```code
bash /scripts/smoke/train_base_control.sh
```
inference:
```code
bash /scripts/smoke/inf_base_control.sh
```

## Related Projects
* [SafeDiffCon](https://github.com/AI4Science-WestlakeU/safediffcon) (ICML 2025): We propose safe diffusion models for PDE Control, which introduces the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases.

* [CL-DiffPhyCon](https://github.com/AI4Science-WestlakeU/CL_DiffPhyCon) (ICLR 2025): We introduce an improved, closed-loop version of DiffPhyCon. It has an asynchronous denoising schedule for physical systems control tasks and achieves closed-loop control with significant speedup of sampling efficiency.

* [DiffPhyCon](https://github.com/AI4Science-WestlakeU/diffphycon) (NeurIPS 2024): We introduce DiffPhyCon which uses diffusion generative models to jointly model control and simulation of complex physical systems as a single task.

* [CinDM](https://github.com/AI4Science-WestlakeU/cindm) (ICLR 2024 spotlight): We introduce a method that uses compositional generative models to design boundaries and initial states significantly more complex than the ones seen in training for physical simulations.

## Citation
If you find our work and/or our code useful, please cite us via:

```bibtex
@inproceedings{
  hu2025wavelet,
  title={Wavelet Diffusion Neural Operator},
  author={Peiyan Hu and Rui Wang and Xiang Zheng and Tao Zhang and Haodong Feng and Ruiqi Feng and Long Wei and Yue Wang and Zhi-Ming Ma and Tailin Wu},
  booktitle={The Thirteenth International Conference on Learning Representations},
  year={2025},
  url={https://openreview.net/forum?id=FQhDIGuaJ4}
}
```

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