LinPDE-GP

Physics-Informed Gaussian Process Regression for Linear PDE Solvers

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This fork builds on the original linpde-gp framework in two directions.

Research extensions:

• Helmholtz equation support, including the Helmholtz operator and related components
• An application to in vivo brain magnetic resonance elastography (MRE)
• Additional research modules are in preparation...

Maintenance and reproducibility:

• Updates to the software and submodules for modern dependencies
• A maintained Docker build

The original framework was developed by the author(s) of "Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers." If you use this software in your research, we strongly recommend also visiting the original repository.

Projects

Each project is self-contained and has its own README with the science, the data it needs, and exact reproduce steps. Start at the front door here (install once), then follow the relevant guide.

Project	Status	Guide
Helmholtz equation & Brain MRE — physics-informed GP solver for the inhomogeneous Helmholtz BVP (real & complex, 1D/2D/3D) with FDM/PINN baselines, and its application to in vivo brain magnetic resonance elastography	Open-source	docs/HELMHOLTZ.md

Submodules

This project depends on two Git submodules:

Submodule	Description
probnum	Probabilistic numerics library (forked for compatibility)

Both are fetched automatically when you clone with --recurse-submodules (see below).

Getting Started

Usage (Docker)

Prerequisites: Install Docker (with Docker Compose). On macOS/Windows, Docker Desktop already includes Compose.

Clone with submodules (if you haven't already):

git clone --recurse-submodules https://github.com/bydeng01/linpde-gp.git
cd linpde-gp

If you've already cloned it without submodules, you can fetch them afterward by doing:

cd linpde-gp
git submodule update --init --recursive

Quick Start (recommended)

Start the container (builds if needed and runs in the background):

./docker-run.sh run

Open a shell inside the running container:

./docker-run.sh shell

Run tests:

./docker-run.sh test

Start Jupyter Notebook (accessible on your host at http://localhost:8888):

./docker-run.sh jupyter

Tip: If Jupyter asks for a token, show the container logs to find the URL:

docker-compose logs -f linpde-gp

Stop the container:

./docker-run.sh stop

Clean up everything (container, image, and volumes like pip cache):

./docker-run.sh clean

Notes

• Your project directory is mounted into the container at /app, so local code edits are immediately available inside the container.
• Ports 8888 (Jupyter) and 8000 are published by default; adjust docker-compose.yml if you need different ports.

Manual Docker Compose equivalents

# Build
docker-compose build

# Run (detached)
docker-compose up -d

# Shell
docker-compose exec linpde-gp /bin/bash

# Stop
docker-compose down

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Usage (Local)

If you prefer to work without Docker, install the submodules as editable packages after cloning:

pip install -e ./probnum
pip install -e ./gp_constr
pip install -e .

Citation

If you use or refer to the Helmholtz equation module or the brain MRE application in this fork, please cite our paper.

@misc{deng2026operatorinformedgaussianprocessescomplex,
      title={Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography}, 
      author={Boyuan Deng and Kshitiz Upadhyay and Michael Shields},
      year={2026},
      eprint={2607.14193},
      archivePrefix={arXiv},
      primaryClass={stat.ML},
      url={https://arxiv.org/abs/2607.14193}, 
}

If you use this software, please cite the original publication:

@misc{Pfoertner2022LinPDEGP,
  author = {Pf\"ortner, Marvin and Steinwart, Ingo and Hennig, Philipp and Wenger, Jonathan},
  title = {Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers},
  year = {2022},
  publisher = {arXiv},
  doi = {10.48550/arxiv.2212.12474},
  url = {https://arxiv.org/abs/2212.12474}
}

License

Released under the MIT License.

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