/TaylorPDENet

Primary LanguagePython

TaylorPDENet

Theory

In this repository the code for the TaylorPDENet can be found. It is based on the estimation of the derivatives contained in the PDE (Partial Differential Equation) with the Taylor Polyinomial.

$$ P_n(x,y) = \sum_{q=0}^Q \sum_{r=0}^{Q-j} \frac{(x-x_0)^j(y-y_0)^k }{q!r!} \cdot \frac{\partial^{(q+r)}f}{\partial x^q \partial y^r}(x_0,y_0) $$

The Taylor polyinomial can be rewritten to form a system of linear equations cotaining the derivatives $d_{(q,r)}$ as unknowns.

$$ f(x_i, y_i)-f(x_0, y_0) = \sum_{q+r\leq Q; q,r\geq 0}\alpha^{(i)}(q, r)d_{(q,r)} $$

Where the $\alpha(q, r)$ are the known coefficients computed by the $x$ and $y$ distance to the point $(x_0, y_0)$, and $f(x,y)-f(x_0,y_0)$ are the differences of the values at the points.

$$ \Delta \mathbf{u} = \mathbf{A}\mathbf{D} $$

The single derivatives can then be computed solving the above mentioned linear system of euations for several points $P_i$.

The TaylorPDENet is able to perfrom symbolic regression and find the coefficients of the underlying advection-diffusion equation and perform prediction of future states of the dynamic system.

Setup

Install the requirements.txt file. We use the pytorch lightning framework for training and testing purposes.

Generate data with pde-generator

Basic Usage pde-generator

  • clone repository
  • Install requirements.txt in pde-generator
  • run mkdir data in TaylorPDENet

The following command generates a PDE of the second order and a starting seed (42).

  • Run twice with different seeds (one for training and one for testing).
  • For training data use --seed 42 --save-name 'example_256'
  • For testing data use --seed 31 --save-name 'example_256_test'
python3 main.py \
--t_max 256 \
--grid-size 256 \
--save-dt 0.5 \
--step-size 0.001 \
from_string \
--equation "1.5*u_x_1_y_0^1+1.65*u_x_0_y_1^1+0.55*u_x_2_y_0^1+0.7*u_x_0_y_2^1+1.0*u_y_1_x_1^1" \
--seed 42 \
--save-name 'example_256'

Usage

For training execute

python3 main.py

with

--net -> 'TP_net' (ours)
--equation -> your equation e.g. 'example_256'
--epochs -> number of epochs
--neighbors -> number of neighbors
--gridsize -> number of points on one side; If downsampled overall number of points will be (gridsize * gridsize)
--sample -> Downsample?
--forecast -> number of dt-Blocks
--order -> order of PDE (recommended 2)
--dt -> timestep
--test_forecast -> number of dt-Blocks for testing

Example:

python3 main.py --net 'TP_net' --equation 'example_256' --sample True

In the background pytorch lightning will train the model and finally test the the model after the indicated number of training-epochs. In the end the coefficients are printed in the console.