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PyTorch implementation forPhysics-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 inrequirements.txt and clone our repository

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

Input dataset includes Gaussian random field (GRF) with different truncations ofN leading terms of Karhunen-Loeve expansion (KLE)GRF KLE-N,Warped GRF, andChannelized 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

Change64 to32 to download the dataset with 32x32 grid (mainly for probabilistic surrogate).The dataset is saved at./datasets/.

Train aphysics-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 inParser class.
• --cuda n selectn-th GPU card
• Checkdarcy.py for the PDE loss and boundary loss for the Darcy flow problem.
• Checkimage_gradient.py for Sobel filter to estimate spatial gradients.
• The experiments are saved at./experiments/codec/mixed_residual/.

Train adata-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, seeParser class intrain_codec_mixed_residual.py andtrain_codec_max_likelihood.py.
• The experiments are saved at./experiments/codec/max_likelihood/.

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
• Checkglow_msc.py for the multiscale conditional Glow model
• Use--data-init to speed up training with one minibatch of labeled data.Note that this isnot necessary.
• The experiments are saved in./experiments/cglow/reverse_kl/.

Try more difficult case ofKLE512 over64x64 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.

Convolution Decoder Networks VS Fully-Connected Neural Networks

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

wherealpha1 andalpha2 are the coefficients in the nonlinear constitutive equation.Checkmain() for other hyperparameters insolve_conv_mixed_residual.py.

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.

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.

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}}