基于深度学习近似反射倒向随机偏微分方程

Deep learning based approximate reflection backward stochastic partial differential equations

  • 摘要: 本文将近似一类高维反射倒向随机偏微分方程的解。主要困难是设计倒向离散格式和处理由反射项带来的奇异性。基于Christian Beck等1提出的深度分离方法,我们引入和验证三种近似这类高维反射倒向随机偏微分方程解的算法,包括一种直接近似和两种惩罚近似方法。数值结果表明,在较短的运算时间内,该三种近似方法在50个维度内的近似结果表现稳定,相对误差的精度最小达到了10−3

     

    Abstract: This article will approximate the solution of a class of high-dimensional reflection backward stochastic partial differential equations. The main difficulties are designing a backward discrete format and dealing with the singularity caused by the reflection term. Based on the deep separation method proposed by Christian Beck et al.1, we introduce and validate three algorithms that approximate the solutions of high-dimensional reflection backward stochastic partial differential equations, including one direct approximation and two penalty approximation methods. The numerical results indicate that within a relatively short operating time, the three approximation methods exhibit stable approximation results across 50 dimensions, with their relative error accuracy reaching a minimum of 10−3.

     

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