DrMZ

This package, Deep renormalized Mori-Zwanzig (DrMZ) contains the various submodules for generating data and training operator neural networks and constructing full and reduced order models based on the trained operator neural networks (full and reduced model work is pending).

This package is in a pre-release state with each update potentially breaking. The module is set up to run on CPUs using double precision and during training will spawn (up to) as many threads as there are physical cores available. GPU support is available but needs testing and modification for single precision type consistency.

Adding this package:

This package was recently upgraded to the long-term support (LTS) release of Julia 1.6.7. Download Julia 1.6.7 and check out the platform specific instructions for instructions on adding Julia to PATH. This may have introduced breaking changes which are being reviewed. The original version based on Julia 1.0.5 is available via release tag one.

To add the package from GitHub link using Julia REPL:

] add http_link_to_GitHub_repository

To add the package from direct download from GitHub using Julia REPL:

] add path_to_GitHub_repository_download_location

Accessing the development documentation:

Development documentation complete with working links to the source. Additionally, you can access the documentation from the Julia REPL:

? name_of_function

Running an example operator neural network:

Under the examples directory there is an example script for the advection-diffusion equation on the domain x in [0,2 pi], t in [0,1], and for periodic boundary conditions. The example script, advection_diffusion_generate_train.jl, generates the data set, trains the neural network, and outputs a few results. The data is generated using a 128 mode Fourier expansion with f(sin^2(x/2)) sampled from a Gaussian random field to generate the training and testing initial conditions. To run this script, there are several additional packages that need to be installed/standalone installed in addition to the DrMZ package. To add these packages using Julia REPL:

] add Flux Parameters LaTeXStrings ColorSchemes Plots PyPlot

If you encounter any errors loading all the packages in one go, just add each package sequentially, and it should resolve any issues. Once all the additional packages have been installed, the example script can be run from a terminal session using:

julia advection_diffusion_generate_train.jl

The first time this script is run it may take some time to indicate the dataset is being built as it will show a few warnings and install/build additional required packages. It is recommended to use Julia for Visual Studio Code for running any example scripts that generate and save figures.

References:

The operator neural networks were implemented based on the structure presented by Lu Lu et al. (2020):

Lu, L., Jin, P., & Karniadakis, G. E. (2020). DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193v3.

Details about the custom basis function construction and examples for using the functions to evolve PDEs can be found in:

Meuris, B., Qadeer, S., & Stinis, P. (2023). Machine-learning-based spectral methods for partial differential equations. Scientific Reports, 13(1), 1739.