论文标题
用于高维积分的机器学习求解器：通过随机加权最小化和随机梯度下降来求解Kolmogorov PDES，通过具有Malliavin重量的SDE的高阶弱近似方案
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
论文作者
论文摘要
本文引入了一种非常简单且快速的计算方法，用于高维积分，以求解高维kolmogorov部分微分方程（PDE）。通过用随机梯度下降求解随机加权最小化，获得了新的基于机器学习的方法，该方法的灵感来自具有Malliavin权重的随机微分方程（SDE）的高阶弱近似方案。然后，对高维kolmogorov PDE的解决方案或对高维SDE的溶液功能的期望是准确近似的，而不会遭受维度的诅咒。通过使用二阶和三阶离散化方案显示了PDE和SDE的数值示例最多100个维度，以证明我们方法的有效性。
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.
