Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces

This repository contains the code to replicate the results in the paper: "Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces".

The files rkhs_functions.py and training_functions.py are used to define and train RKHS estimators for the Green's functions and bias terms of fundamental solution operators that appear in many linear PDEs.

Simulated Data (generate_data/)

• karhunen_loeve.py is used to generate all random input forcings, boundary conditions, and initial conditions through the Karhunen-Loeve expansion given an arbitrary kernel function (e.g. squared exponential, periodic squared exponential).
• helmholtz.ipynb generates all input forcings $f(x)$ and solutions $u(x)$ for the 1D Helmholtz equation with wavenumber $\omega$. To simulate 1D Poisson equation use $\omega = 0$.
• schrodinger.ipynb generates all input boundary conditions $b(x)$ and solutions $u(y_1, y_2)$ for the 2D time-independent Schrödinger equation.
• fokker_planck.ipynb generates all initial conditions $u_0(x)$ and solutions $u(y, t)$ for the 1D Fokker-Planck equation. Depends on the simulation toolbox of https://github.com/johnaparker/fplanck which has been modified and saved in the subfolder fplanck/ for our simulation purposes.
• heateq.ipynb generates all forcings $f(x, s)$ and solutions $u(y, t)$ for the 1D heat equation.

Numerical Experiments (experiments/)

• poisson/poisson.ipynb contains the code to learn the Green's function and bias term of the Poisson equation. Used to reproduce Figures 1 & 2 of the paper.
• helmholtz/helmholtz.ipynb learns the Green's function and bias term of the Helmholtz equation and studies the sensitivity of these estimators to samples, measurements, and noise which is used to generate Figure 3.
• schrodinger/schrodinger.ipynb learns the Green's function of the Schrödinger equations from boundary condition to solution which reproduces Figures 4 & 8.
• fokker_planck/fokker_planck.ipynb is used to learn the Green's function of the Fokker-Planck equation from initial condition to solution which reproduces Figures 5 & 9.
• heateq/heateq.ipynb uses time-invariance and time-causal physical constraints to learn the Green's function of the heat equation from forcing to solution. This is used to generate Figures 6 & 10 of the paper.

RKHS Hyperparameter Sweeps (experiments/hyperparam_sweeps)

• This directory contains all the code necessary to reproduce the hyperparameter sweeps for the RKHS kernel lengthscale vs. grid size on the Poisson and Helmholtz equations shown in Figure 7 of the paper.
• These sweeps are time consuming as they iterate over several choices of RKHS kernels, kernel lengthscales, and grid sizes so these experiments were executed on the MIT Supercloud SLURM cluster.
• The shell script sweep_loop.sh subdivides the list of all possible hyperparameter combinations into batches and it test each batch of hyperparameter combinations by calling run_sweep.sh with the sbatch command.
• Each batch of hyperparameter combinations is then tested on the cloud and the results of the fit for each batch are then saved in a text output file of the form sweep_results-n-N where n is the current batch number and N is the total number of batches.
• The outputs of all files can then be collected and saved in an npz file by running save_sweep.py.
• The notebook sweep_results.ipynb then reads in the saved poisson_sweep.npz and helmholtz_sweep.npz files and reproduces Figure 7 of the paper for hyperparameter sweeps on the Poisson and Helmholtz equation.