A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak

12/22/2020

∙

The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from the curse of dimensionality. Numerical examples for PDEs and SDEs up to 100 dimensions are shown by using second and third-order discretization schemes in order to demonstrate the effectiveness of our method.

READ FULL TEXT

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak

Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver

A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs

The Deep Minimizing Movement Scheme

Stochastic Gradient Descent for Semilinear Elliptic Equations with Uncertainties

A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak

Related Research

Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver

A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs

The Deep Minimizing Movement Scheme

Stochastic Gradient Descent for Semilinear Elliptic Equations with Uncertainties

A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems