Fourier-Transformer for solving Fokker-Planck PDE

Transformer for solving the problem in Stochastic Local Volatility

• Basic Component
  • All the files can be found here
  • The problem is denfined in Stochastic Local Volatility, with SDE of 2 dimensions in space and 1 dimension in time.
• Stages of the lab
  • Experiment_0, sampling of the FokkerPlanck PDE, with the help of Quantlib.
  • Experiment_1, FNO in solving the FokkerPlanck PDE.
  • Experiment_2, FNO Transformer in solving the FokkerPlanck PDE.
• Basic Introduction
  • There are also some RL based methods to solve it
    • In solving HJB PDE, Actor-Critic Method for High Dimensional Static Hamilton-Jacobi-Bellman Partial Differential Equations based on Neural Networks
    • In solving Elliptic PDE, Neural Q-learning for solving elliptic PDEs

Week4

2022.May.09, 11:00AM~12:15Noon, with Dr. Shuhao Cao, Ziheng Wang, and Deqing Jiang at Mathematical Institute.

• The interface bwtween Quantlib and Transform
  • Sampling point format
  • Pre-process data format on our own
• The result of the PDE is in a stochastic way, which is totally different from the
• 
  In order to get v, the p is what we want to solve, the transition density. It can be discribed by The Fokker-Planck Equation, which is 2 dimensions in space and 1 dimension in time.
  
• #TODO Problems still remains
  • How to define the input data format
  • How to define the final result of our model
  • Is there any advantage compared with the tradition method Monte Carlo?
• Reference paper
  • Fourier Neural Operator Fourier Neural Operator for Parametric Partial Differential Equations
  • Galerkin Transformer Choose a Transformer Fourier or Galerkin
  • Fourier Transformer Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers
  • DeepXDE A deep learning library for solving differential equations Documentation

Week3

• Problems remian in this week
  • Only a set of PDEs are suitable for Fourier Transform.
  • Why sample is 256*256 and downsample is 64*64, what has downsampling done here?
  • How MCMC using FNO？
  • To solve different PDEs, do we need to change the code in the FNO?
    • Or the info has been included in the dataset by sampling.
  • We need to solve a two dimensional difussion equation.
    • https://github.com/KangOxford/Fourier-Transformer/blob/6fd306e6a6856a46b3db78fa57db8ced3c307060/fourier_neural_operator/fourier_3d.py#L1-L7
    • In the class SpectralConv3d(nn.Module):
      https://github.com/KangOxford/Fourier-Transformer/blob/90686ea11466e55739068a4ca5b56fa74063d6f5/fourier_neural_operator/fourier_3d.py#L61-L76
  • How does the advantage of operator learning functions in this lab?
    • Does it not need to train again with different intial condition or boundrary condition.
    • Perhaps. As the Operator is learnt. So one of the three condtion has been solved, with the other two to be the intial condition and boundrary condition.
• Try with FNO for basic set up of the experiment.
• The FNO model for solving entire family of PDEs.
  • FNO project
  • FNO GitHub
  • FNO introduction
  • FNO data
  • FNO models
    • Here are the pre-trained models. It can be evaluated using eval.py or super_resolution.py
• By now there is no need to focus on the Galerkin Transformer, as it only provides a better precision on results.
• We will focus more on the FNO in the next stages.
• The FNO is more focused on solving a family of PDEs without try to train the neural network from the scratch, such as when the initial condition or the boundary condition change.