Physics-Constrained Surrogates without Labeled Data

PyTorch implementation for Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. This is accomplished by appropriately incorporating the governing equations into the loss / likelihood functions, as demonstrated with both deterministic surrogates (convolutional encoder-decoder networks) and probabilistic surrogates (flow-based conditional generative models).

Yinhao Zhu, Nicholas Zabaras, Phaedon-Stelios Koutsourelakis, Paris Perdikaris

Codec - GRF KLE512	Codec - Channelized	cGlow - GRF KLE100

Install dependencies in requirements.txt and clone our repository

git clone https://github.com/cics-nd/pde-surrogate.git
cd pde-surrogate

Dataset

Input dataset includes Gaussian random field (GRF) with different truncations of N leading terms of Karhunen-Loeve expansion (KLE) GRF KLE-N, Warped GRF, and Channelized field. Corresponding output are solved with FEniCS. Note that only input samples are needed to train the physics-constrained surrogates.

Download the dataset with 64x64 grid

bash ./scripts/download_datasets.sh 64

Change 64 to 32 to download the dataset with 32x32 grid (mainly for probabilistic surrogate). The dataset is saved at ./datasets/.

Deterministic Surrogates - Convolutional Encoder-Decoder Networks

Physics-constrained surrogates

Train a physics-constrained surrogate with mixed residual loss without output data

python train_codec_mixed_residual.py --data grf_kle512 --ntrain 4096 --batch-size 32

• Use --data channelized to train the surrogate for channelized permeability fields.
• Choose smaller --batch-size when num of training data --ntrain is smaller, check more hyperparameters in Parser class.
• --cuda n select n-th GPU card
• Check darcy.py for the PDE loss and boundary loss for the Darcy flow problem.
• Check image_gradient.py for Sobel filter to estimate spatial gradients.
• The experiments are saved at ./experiments/codec/mixed_residual/.

Data-driven surrogates

Train a data-driven surrogate with maximum likelihood, which requires output data

python train_codec_max_likelihood.py --data grf_kle512 --ntrain 4096 --batch-size 32

• You may try different --data, --ntrain, --batch-size, and many other hyperparameters, see Parser class in train_codec_mixed_residual.py and train_codec_max_likelihood.py.
• The experiments are saved at ./experiments/codec/max_likelihood/.

Probabilistic Surrogates - Conditional Glow

Train conditional Glow with reverse KL divergence loss without output data

python train_cglow_reverse_kl.py --beta 150 --ntrain 4096 --kle 100 --imsize 32

Tune the network structure by setting hyperparameters, e.g.

• --beta: precision parameter for the reference density
• --enc-blocks: e.g. [3, 4, 4], a list of # layers in each dense block of encoder network
• --flow-blocks: e.g. [6, 6, 6], a list of # steps of flow in each level of the Glow model
• --coupling: 'dense' or 'wide', the type of coupling network for affine coupling layer
• Check glow_msc.py for the multiscale conditional Glow model
• Use --data-init to speed up training with one minibatch of labeled data. Note that this is not necessary.
• The experiments are saved in ./experiments/cglow/reverse_kl/.

Try more difficult case of KLE512 over 64x64 grid

python train_cglow_reverse_kl.py --beta 150 --ntrain 8192 --kle 512 --imsize 64 --lr 0.001

Also modify --enc-blocks to be [3, 3, 3, 3], and --flow-blocks to be [4, 4, 4, 4]. You should expect much longer training time and potentially unstable training. When it is unstable, use additional --resume to resume training from the latest checkpoint saved, or --ckpt-epoch 100 to resume from specific checkpoint, e.g. at 100 epochs. Use --data-init also helps.

Solving Deterministic PDEs

Convolution Decoder Networks VS Fully-Connected Neural Networks

To solve Darcy flow equation with ConvNet

python solve_conv_mixed_residual.py --data grf --kle 1024 --idx 8 --verbose

• --data: ['grf', 'channelized', 'warped_grf']
• --kle: [512, 128, 1024, 2048], no need to set for 'channelized' and 'warped_grf'
• --idx: 0-999, index for selecting which input to solve for 'grf', 'warped_grf', 0-512 for 'channelized'
• --versbose: add this flag to show more detailed output in terminal
• Results are saved at ./experiments/solver/

For solving nonlinear PDEs, add --nonlinear flag, which will call FEniCS to solve the nonlinear Darcy flow (fenics.py as the reference for the ConvNet solution.

python solve_conv_mixed_residual.py --data grf --kle 1024 --idx 8 --nonlinear --alpha1 0.1 --alpha2 0.1

where alpha1 and alpha2 are the coefficients in the nonlinear constitutive equation. Check main() for other hyperparameters in solve_conv_mixed_residual.py.

To solve Darcy flow with fully-connected neural nets

python solve_fc_mixed_residual.py --data grf --kle 512 --idx 8 --verbose

Same hyperparameters as the ConvNet case. Nonlinear PDE case is not investigated here.

Pretrained Models

Download the pre-trained probabilistic surrogates

bash ./scripts/download_checkpoints.sh

Then you can check useful post-processing functions, including the ones for uncertainty quantification.

python post_cglow.py

• Use --run-dir to specify the directory for your own runs, default is the downloaded pretrained model.

Citation

If you use this code for your research, please cite our paper.

@article{zhu2019physics,
  title={Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data},
  author={Yinhao Zhu and Nicholas Zabaras and Phaedon-Stelios Koutsourelakis and Paris Perdikaris},
  journal={Journal of Computational Physics},
  volume = "394",
  pages = "56 - 81",
  year={2019},
  issn={0021-9991},
  doi={https://doi.org/10.1016/j.jcp.2019.05.024}
}