Deep BSDE Solver in TensorFlow (2.0)

Quick Installation

For a quick installation, you can create a conda environment for Python using the following command:

conda env create -f environment.yml

Training

python main.py --config_path=configs/hjb_lq_d100.json

Command-line flags:

• config_path: Config path corresponding to the partial differential equation (PDE) to solve. There are seven PDEs implemented so far. See Problems section below.
• exp_name: Name of numerical experiment, prefix of logging and output.
• log_dir: Directory to write logging and output array.

Problems

equation.py and config.py now support the following problems:

Three examples in ref [1]:

• HJBLQ: Hamilton-Jacobi-Bellman (HJB) equation.
• AllenCahn: Allen-Cahn equation with a cubic nonlinearity.
• PricingDefaultRisk: Nonlinear Black-Scholes equation with default risk in consideration.

Four examples in ref [2]:

• PricingDiffRate: Nonlinear Black-Scholes equation for the pricing of European financial derivatives with different interest rates for borrowing and lending.
• BurgersType: Multidimensional Burgers-type PDEs with explicit solution.
• QuadraticGradient: An example PDE with quadratically growing derivatives and an explicit solution.
• ReactionDiffusion: Time-dependent reaction-diffusion-type example PDE with oscillating explicit solutions.

New problems can be added very easily. Inherit the class equation in equation.py and define the new problem. Note that the generator function and terminal function should be TensorFlow operations while the sample function can be python operation. A proper config is needed as well.

Dependencies

Please be aware that the code may not be compatible with the latest version of TensorFlow.

For those using older versions, a version of the deep BSDE solver that is compatible with TensorFlow 1.12 and Python 2 can be found in commit 9d4e332.

Reference

[1] Han, J., Jentzen, A., and E, W. Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning, Proceedings of the National Academy of Sciences, 115(34), 8505-8510 (2018). [journal] [arXiv]
[2] E, W., Han, J., and Jentzen, A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5, 349–380 (2017). [journal] [arXiv]

Citation

@article{HanArnulfE2018solving,
  title={Solving high-dimensional partial differential equations using deep learning},
  author={Han, Jiequn and Jentzen, Arnulf and E, Weinan},
  journal={Proceedings of the National Academy of Sciences},
  volume={115},
  number={34},
  pages={8505--8510},
  year={2018},
  publisher={National Acad Sciences},
  url={https://doi.org/10.1073/pnas.1718942115}
}

@article{EHanArnulf2017deep,
  author={E, Weinan and Han, Jiequn and Jentzen, Arnulf},
  title={Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations},
  journal={Communications in mathematics and statistics},
  volume={5},
  number={4},
  pages={349--380},
  year={2017},
  publisher={Springer},
  url={https://doi.org/10.1007/s40304-017-0117-6}
}