In-Context Operator Networks for Sturm-Liouville problems

This repo contains an implementation of ICON (In-Context Operator Networks) on the Sturm-Liouville problem to find the ground state eigenvectors (eigenvectors corresponding to the smallest eigenvalue). The conditions and QoIs (quantities of interest) will be the potential function $q(x)$ and $\lambda_0$, respectively with the $p$ and $w$ being the identity function

Once in the directory containing the contents of the repository, run

pip install -r requirements.txt

if you do not have all the necessary packages listed. Or, if you are using conda, you use the commands

conda env create --file environment.yml
conda activate icon

and

conda env export > environment.yml 

and

conda list --export > requirements.txt

to update the environment whenever there's been a change made in the dependencies.

More on conda with GPUS:

• https://fmorenovr.medium.com/set-up-conda-environment-pytorch-1-7-cuda-11-1-96a8e93014cc

Sturm-Liouville Problem

Given a differential equation of the form:

$$ -\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u = \lambda w(x)u $$

subject to certain boundary conditions, where:

• $u(x)$ is the unknown function,
• $p(x)$, $q(x)$, and $w(x)$ are given functions,
• $\lambda$ is an eigenvalue parameter.

The boundary conditions are typically specified as:

1. Homogeneous Dirichlet Boundary Conditions: $u(a) = u(b) = 0$
2. Homogeneous Neumann Boundary Conditions: $u'(a) = u'(b) = 0$
3. Mixed Boundary Conditions: $u(a) = 0, , u'(b) = 0$ or $u'(a) = 0, , u(b) = 0$

The goal is to find the eigenvalues $\lambda_n$ and corresponding eigenfunctions $u_n(x)$ that satisfy the differential equation and the specified boundary conditions.

Instructions

First, create the dataset using the command

python3 dataset/create_dataset.py

which will create a tfrecord in a folder named data. This command also takes in three arguments: number, gridsize, and path. The number flag represents the number of operators to generate, the gridsize flag represents the number of points to be sampled in each operator, and the path flag is where the tfrecord dataset will be created. An example command is

python3 dataset/create_dataset.py --number 1000 --gridsize 1001 --path ./data

To train the model, run the following command in the terminal:

python tools/trainer.py

and to test the model,

python tools/predictor.py

To observe tensorboard logs if enabled, use the following command

tensorboard --logdir ./lightning_logs/{current version}

Docker

Run a docker container from a docker image built from the Dockerfile

docker build -t icon .

and then run a container using this command

docker run --name icon --gpus all -it --rm icon

Used these resources to help make dockerfile

• https://saturncloud.io/blog/how-to-install-pytorch-on-the-gpu-with-docker/
• https://stackoverflow.com/questions/65492490/how-to-conda-install-cuda-enabled-pytorch-in-a-docker-container

Relevant Papers

• https://www.pnas.org/doi/epdf/10.1073/pnas.2310142120
• https://arxiv.org/pdf/2208.01066
• https://proceedings.mlr.press/v202/von-oswald23a/von-oswald23a.pdf