maziarraissi/fbsnns

Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations

Forward-Backward Stochastic Neural Networks

Classical numerical methods for solving partial differential equations suffer from the curse of dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.

For more information, please refer to the following: (https://maziarraissi.github.io/FBSNNs/)

• Raissi, Maziar. "Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations." arXiv preprint arXiv:1804.07010 (2018).

• Video: Forward-Backward Stochastic Neural Networks

• Slides: Forward-Backward Stochastic Neural Networks

Citation

@article{raissi2018forward,
  title={Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations},
  author={Raissi, Maziar},
  journal={arXiv preprint arXiv:1804.07010},
  year={2018}
}

fbsnns's Issues

Update project to run with Tensorflow 2

This project runs with Tensorflow 1.x, which is deprecated and the support to this version will become poor soon. In order to get this project reusable through the scientific community, we should to update the code base to run to an updated version of Tensorflow.

Below we have a message from Google Colab when running a code with Tensorflow 1.x.:

WARNING: Tensorflow 1 is deprecated, and support will be removed on August 1, 2022.
After that, `%tensorflow_version 1.x` will throw an error.

Your notebook should be updated to use Tensorflow 2.
See the guide at https://www.tensorflow.org/guide/migrate#migrate-from-tensorflow-1x-to-tensorflow-2.

Minor bug for one-dimensional problems

As a sanity check, I was trying the network for a one-dimensional problem and noticed you need to add axis=[-1] as an argument to tf.squeeze on line 105 of FBSNN.py. Otherwise, TF squeezes axes 0 and 1 (instead of just axis 2). Here is the updated line:

X1 = X0 + self.mu_tf(t0,X0,Y0,Z0)*(t1-t0) + tf.squeeze(tf.matmul(self.sigma_tf(t0,X0,Y0),tf.expand_dims(W1-W0,-1)), axis=[-1])