基于深度学习的随机偏微分方程数值近似算法

Scalable Gradients for Stochastic Differential Equations

The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differential equation whose solution is the gradient, a memory-efficient algorithm for caching noise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equati…

Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes

A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation. The backward stochastic differential equation and the stochastic differential equation are then approximated by first order finite difference schemes, which results in a first order scheme for the backward doubly stochastic differential equation. Numerical experiments are conducted to illustrate the convergence rate of the proposed scheme.

Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods

In this paper, we consider the equivalence of the $p$th moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is $p$th moment exponentially stable, then any of them is $p$th moment exponentially stable for a sufficiently small step size $h$ and $τ$ under the global Lipschitz assumption on the drift and diffusion coefficients

StPINNs - 深度学习框架用于随机微分方程的近似

StPINNs - Deep learning framework for approximation of stochastic differential equations

本文提出StPINNs框架，利用神经网络近似由莱维噪声驱动的随机微分方程解，提供数学方法解决此类问题。

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, t…

多维随机微分方程正保持数值方案的构造

Construction of positivity preserving numerical schemes for multidimensional stochastic differential equations

本文提出一种多维随机微分方程的正保持数值方案，该方案在均方意义下强收敛于真实解，保留了专业术语和数学符号。