高维元件的机器学习求解器:通过高分级弱近似方案,以马利亚文重量的SDE系统,通过随机加权最小化和随机梯度梯度下降,解决科尔莫戈洛夫PDEs (A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights)

Weight · 随机梯度下降 · 近似 · Machine Learning · 维数灾难 ·

2021 年 1 月 7 日

A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights

翻译：高维元件的机器学习求解器:通过高分级弱近似方案,以马利亚文重量的SDE系统,通过随机加权最小化和随机梯度梯度下降,解决科尔莫戈洛夫PDEs

Riu Naito,Toshihiro Yamada

from arxiv, 11 pages, 2 figures; some typos are corrected; numerical results are added

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.

翻译：本文介绍了一种非常简单和快速的高维元件计算方法,以解决高维科尔莫戈罗夫部分差异方程式(PDEs)问题。新的机器学习方法是通过用随机梯度梯度下降来解决一个随机加权最小化方法获得的。该方法的灵感来自一个具有Malliavin重量的高端随机偏差方程式(SDEs)高端微弱近似近似方案。然后,高维科尔莫戈罗夫 PDEs或高维SDE解决方案功能预期的解决方案的解决方案可以准确估计,而不会受到维度的诅咒。PDEs和SDEs最高至100维的数值示例通过使用二级和三级分解方案展示我们方法的有效性。