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SE-PINN: Solving the Schrödinger Equation via Physics-Informed Machine Learning

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<p align="center">
  <a href="https://se-pinn.github.io/">Website</a> |
  <a href="https://github.com/SE-PINN/SE-PINN">GitHub</a> |
  <a href="https://pypi.org/project/sepinn/">PyPI</a>
</p>

SE-PINN is a physics-informed neural network in PyTorch that solves the Schrödinger equation of quantum mechanics.

Through (a) a custom __loss function__ and (b) a custom __neural-network layer__, SE-PINN is constrained to predict quantum-mechanical states that satisfy the mathematical-physical properties of __symmetry__, __normality__, __orthogonality__, and __consistency__. In addition, the model learns not through supervised learning on labeled data but through __unsupervised learning__ via feedback from the Schrödinger equation itself.

SE-PINN was developed at Vanderbilt University by Tiger Du and Alexander Ahrens under the supervision of Prof. Ipek Oguz (https://engineering.vanderbilt.edu/bio/ipek-oguz/).

The design of SE-PINN is based on https://arxiv.org/abs/2203.00451 and https://arxiv.org/abs/1904.08991.

In addition, SE-PINN supports the following features:
- L-BFGS optimization
- A class that supports deterministic training, checkpointing of training, and visualization
- A Python package on PyPI with CI/CD (`pip install sepinn`)
- Documentation as a [website](https://se-pinn.github.io/) and a [Google Colab notebook](https://colab.research.google.com/github/SE-PINN/SE-PINN/blob/main/docs/quantum_harmonic_oscillator.ipynb
)

<h2 align="center">Example</h2>

__Figure 1__ and __Figure 2__ are both visualizations of the ground state (_left_) and the energy of the ground state (_right_) that are predicted by SE-PINN as it trains. The physical system of interest here is the __quantum harmonic oscillator__, which can be used to model diatomic molecules such as diatomic nitrogen, diatomic oxygen, and the hydrogen halides.

As visualized in __Figure 2__, constraining SE-PINN via a special architectural layer — a __hub layer__ — to predict a ground state that satisfies exact symmetry improves its convergence to both the correct ground state and the correct energy.

| **Figure 1**: SE-PINN without Enforcement of Symmetry |
| --- |
| <img src=https://raw.githubusercontent.com/SE-PINN/SE-PINN/main/assets/no_enforcement_of_symmetry.gif> |

| **Figure 2**: SE-PINN with Enforcement of Symmetry |
| --- |
| <img src=https://raw.githubusercontent.com/SE-PINN/SE-PINN/main/assets/enforcement_of_symmetry.gif> |

<h2 align="center">Usage</h2>

__1. Install SE-PINN.__

```
pip install sepinn
```

__2. Import SE-PINN.__

```python
from sepinn.wrappedpinn import WrappedPINN

model = WrappedPINN(...)

model.train(...)
```

<h2 align="center">Documentation</h2>

A Jupyter notebook is available for reference in the `docs` folder as well as through Google Colab and nbviewer.

__Google Colab__ (_Interactive_):

https://colab.research.google.com/github/SE-PINN/SE-PINN/blob/main/docs/quantum_harmonic_oscillator.ipynb

__nbviewer__ (_Non-interactive_):

https://nbviewer.org/github/SE-PINN/SE-PINN/blob/main/docs/quantum_harmonic_oscillator.ipynb

<h2 align="center">Citation</h2>

```
@techreport{DuAhrensOguz2023,
  author={Du, Tiger and Ahrens, Alexander and Oguz, Ipek},
  institution={Vanderbilt University},
  title={Solving the Schrodinger Equation via Physics-Informed Machine Learning},
  year={2023}
}
```

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