论文标题
耦合前回向SPDE的数值近似值
Numerical Approximations of Coupled Forward-Backward SPDEs
论文作者
论文摘要
我们提出并研究了一种将有限元方法和机器学习技术结合的方案，用于耦合非线性前向后偏移的随机偏微分方程（FBSPDE）与均匀的迪里奇特边界条件的数值近似值。确切地说，我们概括了Dunst和Prohl的开创性工作[Siam J. Sci。 Comp。，38（2017），2725--2755]，通过考虑具有更具包容耦合的一般非线性和非本地FBSPDE；提供了独立的证据，并采用了由此产生的有限维方程的不同数值技术。对于此类FBSPD，我们首先证明了强溶液和弱解决方案的存在和独特性。然后，空间结构域中的有限元方法导致通过使用一些基于深度学习的方案来计算有限的前向后随机微分方程（FBSDES），从而导致FBSPDE的近似值。解决了FBSPDE的空间离散化的收敛分析，包括脱钩和耦合的情况在内的数值示例表明我们的方法非常有效。
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725--2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite-dimensional equations are adopted. For such FBSPDEs, we first prove the existence and uniqueness of the strong solution as well as of the weak solution. Then the finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward-backward stochastic differential equations (FBSDEs) which are numerically computed by using some deep learning-based schemes. The convergence analysis is addressed for the spatial discretization of FBSPDEs, and the numerical examples, including both decoupled and coupled cases, indicate that our methods are quite efficient.
