scholarly journals Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

2017 ◽  
Vol 5 (4) ◽  
pp. 349-380 ◽  
Author(s):  
Weinan E ◽  
Jiequn Han ◽  
Arnulf Jentzen
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
High Dimensional ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential

An acceleration scheme for deep learning-based BSDE solver using weak expansions

2020 ◽  
Vol 07 (02) ◽  
pp. 2050012
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Nonlinear Partial Differential Equations ◽  
High Dimensional ◽  
Parabolic Partial Differential Equations ◽  
Weak Approximation ◽  
Learning Method ◽  
Partial Differential

This paper gives an acceleration scheme for deep backward stochastic differential equation (BSDE) solver, a deep learning method for solving BSDEs introduced in Weinan et al. [Weinan, E, J Han and A Jentzen (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5(4), 349–380]. The solutions of nonlinear partial differential equations are quickly estimated using technique of weak approximation even if the dimension is high. In particular, the loss function and the relative error for the target solution become sufficiently small through a smaller number of iteration steps in the new deep BSDE solver.

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

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