DeltaPhi

Code for "DeltaPhi: Learning Physical Trajectory Residual for PDE Solving".

Installing Environment

Create and activate an Anaconda Environment:

conda create -n DeltaPhi python=3.9
conda activate DeltaPhi

Install Pytorch with following commands (refer to Other Versions for additional cuda versions.):

# for CUDA 11.8
conda install pytorch==2.3.0 torchvision==0.18.0 torchaudio==2.3.0 pytorch-cuda=11.8 -c pytorch -c nvidia
# for CUDA 12.1
conda install pytorch==2.3.0 torchvision==0.18.0 torchaudio==2.3.0 pytorch-cuda=12.1 -c pytorch -c nvidia

Install additional required packages with following commands:

pip install -r requirement.txt

Running Experiments

Case 1: Composites

Download the dataset from Google Driver. Put Composites.zip in datasets/ and unzip it.

cd datasets
unzip Composites.zip

The directory structure is as follows:

datasets/Composites/Composites_LBO_basis/Composites_LBO_basis.mat
datasets/Composites/Data/Composites.mat

Run the experiment using the following scripts:

cd Composites

# The code for NORM (Previous):
CUDA_VISIBLE_DEVICES=0 python norm.py 
# The code for NORM (DeltaPhi): 
CUDA_VISIBLE_DEVICES=0 python norm_DeltaPhi.py 

Case 2: HeatTransfer

Download the dataset from Google Driver. Put HeatTransfer.zip in datasets/ and unzip it.

cd datasets
unzip HeatTransfer.zip

The directory structure is as follows:

datasets/HeatTransfer/Data/HeatTransfer.mat
datasets/HeatTransfer/HeatTransfer_LBO_basis/lbe_ev_input.mat
datasets/HeatTransfer/HeatTransfer_LBO_basis/lbe_ev_output.mat

Run the experiment using the following scripts:

cd HeatTransfer

# The code for NORM (Previous):
CUDA_VISIBLE_DEVICES=0 python norm.py 
# The code for NORM (DeltaPhi): 
CUDA_VISIBLE_DEVICES=0 python norm_DeltaPhi.py 

More Cases

Codes for additional simulation cases will be released later.

Create More Residual Neural Operators

You can create a residual neural operator based on an existing direct neural operator by modifying the original Dataset and Model class. All other configurations keep unchanged.

1. Dataset. Implement the Dataset class in which the __getitem__() function returns not only the original input-output function $(a_i,u_i)$, but also the randomly sampled auxiliary sample $(a_{k_i}, u_{k_i}, score_{k_i})$.

2. Model. Modify the Model class in two aspects: (a) Auxiliary sample integration: Concatenate original inputs with auxiliary sample $(a_{k_i}, u_{k_i}, score_{k_i})$. (b) Trajectory residual connection: Take the summation of the auxiliary output function $u_{k_i}$ and original model outputs, as the final model outputs.

Acknowledgement

We greatly appreciate the following open-source works, as they form the foundation of our work.

• https://github.com/gengxiangc/NORM
• https://github.com/neuraloperator/neuraloperator
• https://github.com/HaoZhongkai/GNOT
• https://github.com/alasdairtran/fourierflow
• https://github.com/pdearena/pdearena
• https://github.com/scaomath/galerkin-transformer
• https://github.com/lu-group/mionet