Unsupervised Neural Networks for Solving Differential Equations

This repository implements DEQGAN, a generative adversarial network method for solving ordinary and partial differential equations. Our paper appeared at the AI4Science workshop at ICML 2022.

Minimal Installation

Using Anaconda:

1. conda env create -f environment_minimal.yml
2. conda activate denn_minimal
3. python setup.py develop

The minimal installation includes the dependencies required to reproduce the results of experiments. Additional dependencies include:

• ray for hyperparameter tuning (ray_tune.py)
• plotly for parallel plots
• fenics for finite element methods

The full list of dependencies can be installed via the environment.yml file.

Reproducing Experimental Results

Substitute {key} with the appropriate problem key (e.g. exp, sho, nlo, etc.) and follow instructions for each method below. See the table below for the full list of problem keys.

DEQGAN:

• python denn/experiments.py --pkey {key} --gan

L1 / L2 / Huber:

• Define PyTorch loss in denn/config/{key}.yaml under training.loss_fn (L1=L1Loss, L2=MSELoss, Huber=SmoothL1Loss)
• python denn/experiments.py --pkey {key}

RK4 / FD:

• python denn/traditional.py --pkey {key}

Summary of Experiments

This table details the currently available differential equations and corresponding problem keys.

Key	Equation	Class	Order	Linear
exp	Exponential Decay	ODE	1st	Yes
sho	Simple Harmonic Oscillator	ODE	2nd	Yes
nlo	Damped Nonlinear Oscillator	ODE	2nd	No
coo	Coupled Oscillators	ODE	1st	Yes
sir	SIR Epidemiological Model	ODE	1st	No
ham	Hamiltonian System	ODE	1st	No
ein	Einstein's Gravity System	ODE	1st	No
pos	Poisson Equation	PDE	2nd	Yes
hea	Heat Equation	PDE	2nd	Yes
wav	Wave Equation	PDE	2nd	Yes
bur	Burgers' Equation	PDE	2nd	No
aca	Allen-Cahn Equation	PDE	2nd	No

Citing DEQGAN

If you would like to reference our work, please use the following BibTeX citation!

@misc{randle2020unsupervisedlearningsolutionsdifferential,
      title={Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks}, 
      author={Dylan Randle and Pavlos Protopapas and David Sondak},
      year={2020},
      eprint={2007.11133},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}
@misc{bullwinkel2022deqgan,
      title={DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networks}, 
      author={Blake Bullwinkel and Dylan Randle and Pavlos Protopapas and David Sondak},
      year={2022},
      eprint={2209.07081},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}