README.md

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[**<p style="text-align: center;">Overview**](#overview) | [**Installation**](#installation) | [**More information**](#more-about-torch-harmonics) | [**Getting started**](#getting-started) | [**Contributors**](#contributors) | [**Cite us**](#cite-us) | [**References</p>**](#references)

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<!-- # spherical harmonic transforms -->

# torch-harmonics

[**Overview**](#overview) | [**Installation**](#installation) | [**More information**](#more-about-torch-harmonics) | [**Getting started**](#getting-started) | [**Contributors**](#contributors) | [**Cite us**](#cite-us) | [**References**](#references)

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## Overview

torch-harmonics is a differentiable implementation of the Spherical Harmonic transform in PyTorch. It was originally implemented to enable Spherical Fourier Neural Operators (SFNO). It uses quadrature rules to compute the projection onto the associated Legendre polynomials and FFTs for the projection onto the harmonic basis. This algorithm tends to outperform others with better asymptotic scaling for most practical purposes.

torch-harmonics uses PyTorch primitives to implement these operations, making it fully differentiable. Moreover, the quadrature can be distributed onto multiple ranks making it spatially distributed.

torch-harmonics has been used to implement a variety of differentiable PDE solvers which generated the animations below. Moreover, it has enabled the development of Spherical Fourier Neural Operators (SFNOs) [1].

<div align="center">
<table border="0" cellspacing="0" cellpadding="0">
    <tr>
        <td><img src="https://media.githubusercontent.com/media/NVIDIA/torch-harmonics/main/images/sfno.gif"  width="240"></td>
        <td><img src="https://media.githubusercontent.com/media/NVIDIA/torch-harmonics/main/images/zonal_jet.gif"  width="240"></td>
        <td><img src="https://media.githubusercontent.com/media/NVIDIA/torch-harmonics/main/images/allen-cahn.gif"  width="240"></td>
    </tr>
<!--     <tr>
        <td style="text-align:center; border-style : hidden!important;">Shallow Water Eqns.</td>
        <td style="text-align:center; border-style : hidden!important;">Ginzburg-Landau Eqn.</td>
        <td style="text-align:center; border-style : hidden!important;">Allen-Cahn Eqn.</td>
    </tr>  -->
</table>
</div>


## Installation
Download directyly from PyPI:

```bash
pip install torch-harmonics
```

Build in your environment using the Python package:

```bash
git clone git@github.com:NVIDIA/torch-harmonics.git
cd torch-harmonics
pip install -e .
```

Alternatively, use the Dockerfile to build your custom container after cloning:

```bash
git clone git@github.com:NVIDIA/torch-harmonics.git
cd torch-harmonics
docker build . -t torch_harmonics
docker run --gpus all -it --rm --ipc=host --ulimit memlock=-1 --ulimit stack=67108864 torch_harmonics
```

## More about torch-harmonics

### Spherical harmonics

The [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are special functions defined on the two-dimensional sphere $S^2$ (embedded in three dimensions). They form an orthonormal basis of the space of square-integrable functions defined on the sphere $L^2(S^2)$ and are comparable to the harmonic functions defined on a circle/torus. The spherical harmonics are defined as

$$
Y_l^m(\theta, \lambda) = \sqrt{\frac{(2l + 1)}{4 \pi} \frac{(l - m)!}{(l + m)!}} P_l^m(\cos \theta)  \exp(im\lambda),
$$

where $\theta$ and $\lambda$ are colatitude and longitude respectively, and $P_l^m$ the normalized, [associated Legendre polynomials](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials).

<div align="center">
<img src="https://media.githubusercontent.com/media/NVIDIA/torch-harmonics/main/images/spherical_harmonics.gif" width="432">
<br>
Spherical harmonics up to degree 5
</div>

### Spherical harmonic transform

The spherical harmonic transform (SHT)

$$
f_l^m = \int_{S^2}  \overline{Y_{l}^{m}}(\theta, \lambda) f(\theta, \lambda) \mathrm{d} \mu(\theta, \lambda)
$$

realizes the projection of a signal $f(\theta, \lambda)$ on $S^2$ onto the spherical harmonics basis. The SHT generalizes the Fourier transform on the sphere. Conversely, a truncated series expansion of a function $f$ can be written in terms of spherical harmonics as

$$
f (\theta, \lambda) = \sum_{m=-M}^{M} \exp(im\lambda) \sum_{l=|m|}^{M} \hat f_l^m  P_l^m (\cos \theta),
$$

where $\hat{f}_l^m$, are the expansion coefficients associated to the mode $m$, $n$.

The implementation of the SHT follows the algorithm as presented in [2]. A direct spherical harmonic transform can be accomplished by a Fourier transform

$$
\hat f^m(\theta) = \frac{1}{2 \pi} \int_{0}^{2\pi} f(\theta, \lambda) \exp(-im\lambda) \mathrm{d} \lambda
$$

in longitude and a Legendre transform

$$
\hat f_l^m = \frac{1}{2} \int^{\pi}_0 \hat f^{m} (\theta) P_l^m (\cos \theta) \sin \theta \mathrm{d} \theta
$$

in latitude.

### Discrete Legendre transform

The second integral, which computed the projection onto the Legendre polynomials is realized with quadrature. On the Gaussian grid, we use Gaussian quadrature in the $\cos \theta$ domain. The integral

$$
\hat f_l^m = \frac{1}{2} \int_{-1}^1 \hat{f}^m(\arccos x) P_l^m (x) \mathrm{d} x
$$

is obtained with the substitution $x = \cos \theta$ and then approximated by the sum

$$
\hat f_l^m = \sum_{j=1}^{N_\theta}  \hat{f}^m(\arccos x_j) P_l^m(x_j) w_j.
$$

Here, $x_j \in [-1,1]$ are the quadrature nodes with the respective quadrature weights $w_j$.

## Getting started

The main functionality of `torch_harmonics` is provided in the form of `torch.nn.Modules` for composability. A minimum example is given by:

```python
import torch
import torch_harmonics as th

device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

nlat = 512
nlon = 2*nlat
batch_size = 32
signal = torch.randn(batch_size, nlat, nlon)

# transform data on an equiangular grid
sht = th.RealSHT(nlat, nlon, grid="equiangular").to(device)

coeffs = sht(signal)
```

To enable scalable model-parallelism, `torch-harmonics` implements a distributed variant of the SHT located in `torch_harmonics.distributed`.

Detailed usage of torch-harmonics is demonstrated in a series of notebooks:

1. [Getting started](./notebooks/getting_started.ipynb)
2. [Quadrature](./notebooks/quadrature.ipynb)
3. [Visualizing the spherical harmonics](./notebooks/plot_spherical_harmonics.ipynb)
4. [Solving the Helmholtz equation](./notebooks/helmholtz.ipynb)
5. [Solving the shallow water equations](./notebooks/shallow_water_equations.ipynb)
6. [Training Spherical Fourier Neural Operators](./notebooks/train_sfno.ipynb)

## Contributors

[Boris Bonev](https://bonevbs.github.io) (bbonev@nvidia.com), [Thorsten Kurth](https://github.com/azrael417) (tkurth@nvidia.com), [Christian Hundt](https://github.com/gravitino) (chundt@nvidia.com), [Nikola Kovachki](https://kovachki.github.io) (nkovachki@nvidia.com), [Jean Kossaifi](http://jeankossaifi.com) (jkossaifi@nvidia.com)  

## Cite us

If you use `torch-harmonics` in an academic paper, please cite [1]

```bibtex
@misc{bonev2023spherical,
      title={Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere}, 
      author={Boris Bonev and Thorsten Kurth and Christian Hundt and Jaideep Pathak and Maximilian Baust and Karthik Kashinath and Anima Anandkumar},
      year={2023},
      eprint={2306.03838},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}
```

## References

<a id="1">[1]</a> 
Bonev B., Kurth T., Hundt C., Pathak, J., Baust M., Kashinath K., Anandkumar A.;
Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere;
arXiv 2306.0383, 2023.

<a id="1">[2]</a> 
Schaeffer N.;
Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations;
G3: Geochemistry, Geophysics, Geosystems, 2013.

<a id="1">[3]</a> 
Wang B., Wang L., Xie Z.;
Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids;
Adv Comput Math, 2018.