Neural Inverse Operators for solving PDE Inverse Problems

This repository is the official implementation of the paper Neural Inverse Operators for solving PDE Inverse Problems

Requirements

The code is based on python 3 (version 3.7) and the packages required can be installed with

python3 -m pip install -r requirements.txt

Source Data

We cover instances of the Poisson, Helmholtz and Radiative Transport equations. Data can be downloaded from https://zenodo.org/record/7566430 (14GB). Alternatively, run the script download_data.py which downloads all required data into the appropriate folder (it requires 'wget' to be installed on your system).

python3 download_data.py

The data for the Seismic Imaging problem can be downloaded at: https://openfwi-lanl.github.io/docs/data.html#vel. Then, the h5 file required to run the code can be built by running: GetStyleData.py and GetCurveData.py

Models Training

Each of the benchmarks described in tha peper can be trained by running the python scripts TrainNio.py. In order to ba able to run the script, the following arguments have to be added (in the order):

• name of the folder where to save the results
• flag for the problem
• flag for the model
• number of workers (usually 0, 1, or 2)

The flag for the problem must be one among:

• sine for the Caldéron problem with trigonometric coefficients
• eit for the Caldéron problem with Heart&Lungs
• helm for the inverse wave scattering
• rad for the radiative transfer problem
• curve for the seismic imaging with the CurveVel-A dataset
• style for the seismic imaging with the CurveVel-A dataset

The flag for the problem must be one among:

• nio_new for NIO
• fcnn for Fully Convolutional NN
• don for DeepONet

For instance:

python3 RunNio.py Example helm nio_new 0

The models' hyperparameter can be specified in the corresponding python scripts as well. To train the InversionNet model (the fully convolutional network baseline for Seismic Imaging) please refer to the GitHub page of Deng et Al (https://arxiv.org/pdf/2111.02926.pdf): https://github.com/lanl/OpenFWI

Hyperparameters Grid/Random Search

Cross validation for each model can be run with:

python3 ModelSelectionNIO.py model which

which and model must be one of the problems and models above. For examples

python3 ModelSelectionNIO.py nio_new helm

For the Seismic Imaging problem, only NIO and DON models can be run.

The hyperparameters of the models in the Table 1 have been obtained in this way.

The best performing configuration can be obtained by visualizing the results with tensorboard:

tensorboard --logdir=NameOfTheModelSelectionFolder

If a SLURM remote server is available set sbatch=True and cluster="true" in the script.

Pretrained Models

The models trained and used to compute the errors in Table 1 can be downloaded (9GB) by running:

python3 download_models.py

Remark: the compressed folder has to be unzipped!

Error Computations

The errors of the best performing models (Table 1) can be computed by running the script ComputeNoiseErrors.py. It will compute the testing error for all the benchmark, for all the models and for different noise levels (Table 6).