The VarMiON tutorial

This is the source code for the VarMiON tutorial. If you have any comments, corrections or questions, please submit an issue in the issue tracker.

Python implementation of a Variationally Mimetic Operator Network for a PDE.

Repository structure

In this repository you can find two floders: "Heat_eq" and "Stokes". In "Heat_eq" there are the files to create and solve VarMiON for heat equation. In "Stokes" there are the files to create and solve VarMiON for the time-dependent Stokes problem.

Generation of PDE data

Here you can find the files to generate the PDE data to train your VarMiON in

• data_generation_equation_to_solve_fenicsx.py : you can generate and save the data by exploiting the numerical solution of the pde with the Python's Library "FEniCSx"; this file requires version 0.9.0 of DOLFINx, you can run a Docker image with DOLFINx with the command docker run -ti dolfinx/dolfinx:v0.9.0
• data_generation_template.py : if you want to use your data, this file shows how to save it in the correct format.

Requirements

To run this project, create a Conda environment with the required packages. Replace environment_name with a name of your choice:

conda create -n environment_name python=3.11 matplotlib scipy seaborn pytorch pytorch-cuda=11.8 -c pytorch -c nvidia

Note: Python and PyTorch must be compatible with the CUDA version installed on your system. For more details, see the official PyTorch installation guide https://pytorch.org/get-started/locally/.

To use the environment in Jupyter notebooks:

source activate base
conda activate environment_name
conda install ipykernel
python -m ipykernel install --user --name environment_name --display-name "Python (environment_name)"

References

Patel, D., Ray, D., Abdelmalik, M.R., Hughes, T.J., Oberai, A.A.: Variationally mimetic operator networks. Computer Methods in Applied Mechanics and Engineering 419 (2024). https://doi.org/10.1016/j.cma.2023.116536

E. Chinellato, P. Martin, L. Rinaldi, and F. Marcuzzi: Exploiting scientific machine learning on embedded digital twins. Springer series: Lectures Notes in Computational Science and Engineering - Math to Product (2025). https://link.springer.com/book/9783031957086.

SPDX-License-Identifier: GPL-3.0
Copyright (c) 2025 NLALDlab
Authors: NLALDlab, Marco Dell'Orto, Laura Rinaldi, Enrico Caregnato, Fabio Marcuzzi