yifanc96 / NonLinPDEs-GPsolver

code for the paper: Solving and Learning Nonlinear PDEs with Gaussian Processes

NonlinearPDEs_GPsolver

Code for the paper Solving and Learning Nonlinear PDEs with Gaussian Processes Solvers with the flavor of automation in scientific machine learning: a general framework for PDEs and inverse problems in science.

Packages required: JAX

• JAX is used for automatic differentiation and vectorization of the constriction of Gram matrices, and the Gauss-Newton iteration. It can be avoid if the users supplement the derivatives manually.
• Note: Codes may plot some figures along with its execution. Please make sure latex is supported in the machine environment, otherwise you may need to change the plot configuration in the code manually.
• Note: the algorithm in the paper https://arxiv.org/abs/2103.12959 can be optimized further to greatly improve the efficiency. Please follow up with our recent progress.
• The current code in this repo uses autodifferentiation and works with dense kernel matrices that can be costly when you use more than 10k collocation points. The repo https://github.com/yifanc96/PDEs-GP-KoleskySolver presents a near linear complexity algorithm for the GP-PDE solver that scale the computation up
• The repo https://github.com/yifanc96/GaussianProcessPDEs.jl presents simple demonstrations of other applications of this methodology
• The folder notebook contains demonstration codes in the Jupyter notebook style, which can be easier to parse and further modify

For demonstration of use in solving PDEs and inverse problems covered in the paper

run the following in your terminal

• python main_NonLinElliptic2d.py --kernel Gaussian --kernel_parameter 0.2 --nugget 1e-13 --N_domain 900 --N_boundary 124 --GNsteps 4 --show_figure True

• python main_Burgers1d.py --kernel anisotropic_Gaussian --kernel_parameter 0.3 0.05 --nugget 1e-5 --N_domain 1000 --N_boundary 200 --GNsteps 8 --show_figure True

• python main_Eikonal2d.py --kernel Gaussian --kernel_parameter 0.2 --nugget 1e-5 --N_domain 1000 --N_boundary 200 --GNsteps 8 --show_figure True

• python main_DarcyFlow2d.py --kernel Gaussian --kernel_parameter 0.2 --nugget 1e-8 --N_domain 400 --N_boundary 100 --N_data 60 --noise_level 0.001 --GNsteps 8 --show_figure True

The architecture of this code

In the src folder:

• PDEs.py contains PDEs classes with built-in methods that are used to run our algorithm, while InverseProblems.py contains the corresponding part for inverse problems
• sample_points.py provides several ways of sampling collocation points; users can also provide their own setting of collocation points
• kernels.py provides a collection of kernel functions and their derivatives
• Gram_matrice.py constructs the kernel matrix used in training and testing stages for our GP based method
• solver.py contains a high level class that integrate the above three files to run the algorithms for any PDEs and inverse problems

In addition, folder reference_solver contains several classical solvers for these PDEs, which are used for comparison purposes