Unified Hard-Constraint Framework

This repository is the official implementation of the NeurIPS 2022 paper: A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs.

Introduction

Physics-informed neural networks are suffering from the unbalanced competition in their loss function: $\mathcal{L}{\mathrm{PDE}}$ vs. $\mathcal{L}{\mathrm{BC}}$. Hard-constraint methods have been developed to alleviate such issues, which are, however, limited to Dirichlet boundary conditions. Our work provides a unified hard-constraint framework for the three most common types of boundary conditions: Dirichlet, Neumann, and Robin. The proposed method has achieved promising performance in real-world geometrically complex problems.

Directory Tree

This repository is organized as:

HardConstraint/
│  README.md
│  requirements.txt       # required dependencies
│
├─data                    # data used in this paper
│      case1_pack.txt     # ground truth for "Simulation of a 2D battery pack (Heat Equation)"
│      case2_airfoil.txt  # ground truth for "Simulation of an Airfoil (Navier-Stokes Equations)"
│      w1015.dat          # achor points of the airfoil
│
├─model/                  # saved model weights (empty)
├─outs/                   # outputs (empty)
└─src
    │  case1.py           # scripts for each experiment 
    │  ...
    │
    ├─configs             # configurations for each experiment 
    │  │
    │  ├─case1
    │  │      ...
    │  │
    │  ├─case2
    │  │      ...
    │  │
    │  └─case3
    │          ...
    │
    ├─FBPINN/             # implementations of each model 
    ├─HC/
    ├─PFNN/
    ├─xPINN/
    │
    └─utils/              # some utils

Getting Started

1. Install necessary dependencies:

   pip install -r requirements.txt

2. To train and evaluate the models in the paper, run this command:

   # In the root directory of this repository
   DDEBACKEND=pytorch python -m src.caseX

   where X = 1, 2, 3, 4, corresponding to Simulation of a 2D battery pack (Heat Equation), Simulation of an Airfoil (Navier-Stokes Equations), High-dimensional Heat Equation, and Ablation Study: Extra fields.

If you want to run different models (i.e., the proposed model and baselines), please modify the global variables in src/caseX.py.

To run the experiment of Ablation Study: Hyper-parameters of Hardness, you can change the value of:

• $\beta_s$ in src/HC/l_functions.py (line 31, default: $\beta_s=5$)
• $\beta_t$ in src/configs/case1/hc.py (line 127, default: $\beta_t=10$, case 1) or src/configs/case3/hc.py (line 56, default: $\beta_t=10$, case 3)

Experimental Results

Settings:

• Evaluation Metrics: Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Weighted Mean Absolute Percentage Error (WMAPE)
• Baselines:
  • PINN: vanilla PINN
  • PINN-LA & PINN-LA-2: PINN with learning rate annealing
  • xPINN & FBPINN: PINN with domain decomposition for geometrically complex PDEs
  • PFNN & PFNN-2: hard-constraint methods based on the variational formulation of PDEs
  • HC: our proposed method

Problems: a 2D battery pack, an airfoil, a high-dimensional heat equation

Results:

Problem & Solution

1. Scalar Type Error

   ...
     File "ENV_PATH/lib/python3.9/site-packages/deepxde/model.py", line 228, in outputs_losses
       outputs_ = self.net(self.net.inputs)
   ...
     File "ENV_PATH/lib/python3.9/site-packages/torch/nn/functional.py", line 1848, in linear
       return torch._C._nn.linear(input, weight, bias)
   RuntimeError: expected scalar type Float but found Double

   Please modify ENV_PATH/lib/python3.9/site-packages/deepxde/model.py (line 228) from:

   self.net.train(mode=training)
   self.net.inputs = torch.as_tensor(inputs)
   self.net.inputs.requires_grad_()
   outputs_ = self.net(self.net.inputs)

   to:

   self.net.train(mode=training)
   self.net.inputs = torch.as_tensor(inputs)
   self.net.inputs.requires_grad_()
   outputs_ = self.net(self.net.inputs.float()) # add this

Citation

If you find this work is helpful for your research, please cite us with the following BibTeX entry:

@article{liu2022unified,
  title={A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs},
  author={Liu, Songming and Hao, Zhongkai and Ying, Chengyang and Su, Hang and Zhu, Jun and Cheng, Ze},
  journal={arXiv preprint arXiv:2210.03526},
  year={2022}
}