unirt/deep-bsde-solver

Solving high-dimensional Partial Differential Equations with Deep Learning

Deep BSDE Solver

This project is part of my master thesis at Imperial College of London. We propose a deep learning approach to solve high-dimensional partial differential equations. The solver is tested on the Black-Scholes Barenblatt equation in 100 dimensions.

Usage

The repository needs to be cloned and running the BlackScholesBarenblatt100D.py will solve the Black-Scholes equation.

$ git clone https://github.com/batuhanguler/Deep-BSDE-Solver.git
$ cd Deep-BSDE-Solver
$ python BlackScholesBarenblatt100D.py

Different architectures are available changing the mode and activation variables in the BlackScholesBarenblatt100D.py file.

Model	mode	activation
Fully-connected with sine activation	"FC"	"Sine"
Fully-connected with ReLU activation	"FC"	"ReLU"
Resnet with sine activation	"Resnet"	"Sine"
Resnet with ReLU activation	"Resnet"	"ReLU"
NAIS-Net with sine activation	"NAIS-Net"	"Sine"
NAIS-Net with ReLU activation	"NAIS-Net"	"ReLU"

Architectures

Fully-connected

A Pytorch version of Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations, a work from Maziar Raissi, is proposed. A simple fully-connected neural network with 5 layers of 256 parameters is implemented.

Resnet Residual networks were proposed in 2015 and helped to backpropagate efficiently the gradient using identity mappings (shortcut connections).

NAIS-Net NAIS-Net were proposed to overcome the problem of forward stability in Residual Networks.

Results