PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers

AAAI 2023 (Accept)

DLDE-II, NeurIPS 2022 Workshop (Spotlight)

[Project page] [Paper-Arxiv] [Paper-Workshop]

Quick Start

1. Installation

Clone PIXEL repo

git clone https://github.com/NamGyuKang/PIXEL.git
cd PIXEL

Create environment

We implemented the 2D, and 3D customized CUDA kernel of the triple backward grid sampler that supports cosine, linear, and smoothstep kernel (Thomas Müller) and third-order gradients $u_{xxc}, u_{yyc}$ with second-order gradients (Tymoteusz Bleja). As a result, the runtime and the memory requirement were significantly reduced. You can find our customized CUDA kernel code at https://github.com/NamGyuKang/CosineSampler.

The code is tested with Python (3.8, 3.9) and PyTorch (1.11, 11.2) with CUDA (>=11.3). You can create an anaconda environment with those requirements by running:

if you use CUDA 11.3, Pytorch 1.11, Python 3.9,
  - conda env create -f CUDA_11_3_Pytorch_1_11_Py_3_9.yml
or with CUDA 11.6, Pytorch 1.12, Python 3.8,
  - conda env create -f CUDA_11_6_Pytorch_1_12_Py_3_8.yml
  
-  conda activate pixel

PIXEL can also be run without CosineSampler, which is a speed-acceleration and memory-efficient interpolation function code using CUDA
(3-order gradient available code). You can run the Python based code of the interpolation function which doesn't use CUDA with the '--cuda-off' command.

2. Run

You can run PIXEL directly using the following code. It is the part of the command.txt file.

[burgers equation]
## Forward problem ##
# Pixel 96 multicell
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --use-cell --n-cells 96 --cell-dim 4 --cell-size 16 --interp cosine --num-train 100000 --num-init 100000 --random-f --f-scale 0.01 --hidden-dim 16 --num-layers 2 --max-iter 39001 --seed 200 --tag sample_tag
# Pixel 64 multicell
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --use-cell --n-cells 64 --cell-dim 4 --cell-size 16 --interp cosine --num-train 100000 --num-init 100000 --random-f --f-scale 0.01 --hidden-dim 16 --num-layers 2 --max-iter 39001 --seed 500 --tag sample_tag
# Pixel 16 multicell
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --use-cell --n-cells 16 --cell-dim 4 --cell-size 16 --interp cosine --num-train 100000 --num-init 100000 --random-f --f-scale 0.01 --hidden-dim 16 --num-layers 2 --max-iter 39001 --seed 500 --tag sample_tag

# PINN
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --num-train 100000 --num-init 100000 --f-scale 0.01 --hidden-dim 40 --num-layers 9 --max-iter 39001 --seed 200 --tag sample_tag

## Inverse problem ##
# PIXEL
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --use-cell --n-cells 192 --cell-dim 4 --cell-size 16 --interp cosine --num-train 100000 --num-init 100000 --random-f --f-scale 0.0005 --hidden-dim 16 --num-layers 2 --max-iter 41 --problem inverse --seed 500 --tag sample_tag
# PINN
python main.py --network base --pde burgers_1d --in-dim 2 --out-dim 1 --num-train 100000 --num-init 100000 --f-scale 0.0005 --hidden-dim 40 --num-layers 9 --max-iter 41 --problem inverse --seed 500 --tag sample_tag


# 3D - Helmholtz
## Forward problem ##
# PIXEL
python main.py --network base --pde helmholtz_3d --a1 7.0 --a2 7.0 --a3 7.0 --in-dim 3 --out-dim 1 --use-cell --n-cells 16 --cell-dim 4 --cell-size 16 --interp cosine --num-train 400000 --num-init 400000 --num-test 100 --random-f --hidden-dim 16 --num-layers 2 --max-iter 11001 --f-scale 0.01 --seed 200 --tag sample_tag 
# PINN
python main.py --network base --pde helmholtz_3d --a1 7.0 --a2 7.0 --a3 7.0 --in-dim 3 --out-dim 1 --num-train 400000 --num-init 400000 --num-test 100 --random-f --hidden-dim 100 --num-layers 8 --max-iter 11001 --f-scale 0.01 --seed 200 --tag sample_tag 


# 3D - Navier-Stokes
## Inverse problem ##
# PIXEL
python main.py --network base --pde navier_stokes_3d --in-dim 3 --out-dim 3 --use-cell --n-cells 150 --cell-dim 4 --cell-size 16 --interp cosine --num-train 100000 --num-init 100000 --random-f --f-scale 1.25 --hidden-dim 16 --num-layers 2 --max-iter 501 --seed 300 --problem inverse --tag sample_tag 
# PINN
python main.py --network base --pde navier_stokes_3d --in-dim 3 --out-dim 3 --num-train 100000 --num-init 100000 --num-test 250 --hidden-dim 20 --num-layers 10 --max-iter 1001 --f-scale 1.25 --seed 300 --problem inverse --tag sample_tag

Citation

If you use this code for research, please consider citing:

@article{kang2023pixel,
title={PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers},
author={Kang, Namgyu and Lee, Byeonghyeon and Hong, Youngjoon and Yun, Seok-Bae and Park, Eunbyung},
journal={Proceedings of the AAAI Conference on Artificial Intelligence}, 
year={2023}}