This repository contains the methods and techniques implemented in the context of the thesis research project of MSc in Computational Science at the University of Amsterdam.
In this work, a non-intrusive parametric surrogate model is presented, able to provide temporal and parametric predictions for systems characterized by high state dimension and nonlinear behavior. In detail, we propose a parametrization algorithm to extend the kernel-based LANDO framework (Baddoo et al., 2022) to a parametric format (parametric LANDO) using deep learning techniques. Additionally, we propose the application of dimensionality reduction techniques to reduce the training cost of parametric LANDO for high-dimensional problems. We showcase the effectiveness of the proposed parametric surrogate model in predicting the parametric dynamics of three numerical problems with nonlinear behavior; namely the Lotka-Volterra model, a 2D heat equation, and the Allen-Cahn equation.
Several experiments are performed, regarding the application of the parametric surrogate model in the three aforementioned numerical problems. Additionally, we perform experiments to investigate whether the performance of parametric LANDO is influenced by the employment of the aforementioned dimensionality reduction techniques.
To perform the experiments regarding the application of parametric LANDO in the Lotka-Volterra model, the following python files should be run:
Experiments_1D.py(The parameter space is one-dimensional)Experiments_2D.py(The parameter space is two-dimensional)Experiment_Cardinality_Training.py(Investigate the dependency of the parametric surrogate model's performance on the cardinality of the training set)
For this numerical problem, the snapshot data for the training of the parametric framework are generated with the finite element method (FEM). The implementation of the finite element simulation is based on the open source software FreeFEM.
To generate the numerical data, please run the Data_Generation.py, Heat_problem_thesis.edp, and Data_Preprocessing.py files in that order.
Subsequently, parametric LANDO can be employed to approximate the parametric dynamics of this system with the Experiments_DiffTimes.py file.
When running this file, several results regarding the performance of parametric LANDO are automatically saved in a .pkl file.
These results can be visualized by running the Visualise_errors.pyfile.
As a last step, the Visualization.edp file can be run to obtain the .vtk visualisations of the reference solutions, predicted solutions, and prediction errors of the parametric surrogate model for several parametric instances.
To generate the snapshot data, run the Data_Generation.py file.
To obtain results from the application of parametric LANDO in this system, run the Experiments_DiffTimes.py, in the folder of the Allen-Cahn equation.
To obtain results regarding the application of POD during the training of parametric LANDO for the Allen-Cahn equation or the heat equation, run the POD_Experiment.py file.
Again, results regarding the performance of parametric LANDO in approximating the dynamics of the Allen-Cahn equation can be visualised by running the Visualise_errors.pyfile, in the Allen-Cahn folder.
- In order to run the aforementioned files, please change the names of the directories in the code, to save the results and figures in your local device.
- Note that FEM simulation for data generation of the heat equation is computationally expensive, so this might take a longer time to run.